Sandi Yonikus: Building the Earth

Liturgical Press  11468 (1968)

This “pre-NOM” album (which means we’ve had one and a half liturgical upheavals since then) is, despite its pretentious title, a mixture of a children’s’ album and early Catholic folk.  Or maybe the pretentious title is reasonable: children are the future, something that the dropping birth rate of the time tended to lose sight of.  In any case, it’s a reasonable effort in both respects.  It’s also a composite effort: in addition to the children from the parish school, it includes seminarians from St. Mary’s Seminary in Houston and some help from the Catholic Student Centre at the University of Texas.  (That’s hard to take for an Aggie, but…we knew how to deal with Catholic students from Austin.)

In addition to this effort, Sandi Yonikus (1936-1988) was also a writer of children’s books.

The songs:

  1. Building The Earth
  2. Our God Is Good
  3. Spirit Of God
  4. Gio (The Little Yellow Bird)
  5. We Come As Your People
  6. I’ll Find Me A Mountain
  7. He’ll Come Again
  8. Knock On Any Door
  9. Sing Alleluia
  10. Teach Me
  11. Christ Takes His Throne
  12. Sing With Joy
  13. Gather ‘Round The Table

Download Building the Earth

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The Last War of the Hippie Dreamers

With the Iowa caucuses in the rear view mirror, our Presidential campaign begins to clarify a bit.  Candidates drop out, others give mixed signals, still others need to quit but haven’t figured it out yet.  Most of that action is on the Republican side, but that’s where most of the candidates are.  When you have about four times as many candidates as you need to have, a lot of dropping out is in order.

The fact that the Republicans have so many candidates is a product of many things, not the least of which is that the Republican Party still believes in the electoral process to effect change while the Democrats do not.  Which one is right is a central question, not only here but in Europe, where the EU has worked hard to insulate itself from popular opinion expressed in elections, with dubious consequences.

Irrespective of their idea of how things get done, the Democrats have to nominate someone to warm the seat in the Oval Office, lest someone else spoil the parade.  The Democrats pride themselves in being the party of the future, demographically, sexually, economically, etc.  In particular the Democrats believe themselves to be the party of the Millennials.  So why has their contest come down to a slugfest between two old white people, while all the blacks, Hispanics, etc., are in the other horse race?

In his book 1973 Nervous Breakdown, Andreas Killen ends with the statement “…the crises of the 1970’s are not so easily buried; indeed they have reemerged with new intensity in our own time.”  To a large extent, the American left defines the “future” as the fulfilment of the revolution that took place in American culture in the 1960’s and 1970’s, and the two Democrats struggling for their party’s nomination represent two different sides of that revolution.

In one corner is Bernie Sanders, who is the “purist” of the two.  It’s time, Bernie tells us, to finish the revolution, to put away the boorish bourgeois, capitalist system that produced the phonies our parents were and come into the economic utopia we’ve always known was out there but were blocked from its entry by another generation of greedy phonies.

In the other corner is Hillary Clinton.  Sometimes it’s hard to tell what she really is, but her career speaks of someone who, like many of her contemporaries, picked themselves off of the floor after the revolution and realised that they’d never get where they wanted to go without working within the system as it was.  That included her marriage to Bill, complete with tolerance for his Scots-Irish penchant for both womanising and not rocking the boat, doing things like balanced budgets and welfare reform.

Bernie for his part has the support of the young, who really are supposed to be the future of both party and country.  The people who support him have been inculcated by a generation of hippie dreamers in the ideals of the 1960’s; the dreamers were better at infiltrating the bureaucracy while their conservative opponents went from election to election.

Hillary for her part has the support of her fellow Boomers and the non-white groups which make up a great deal of the Democrat party’s base.  Most of Bernie’s base–young or old–is white; part of the party’s long term strategy is to ride the increasing non-whiteness of the country to a permanent majority.

At this point, unless Bernie can break out of his white Millennial base, he’s finished; he’s only dragging the process out.  So what do we learn from all of this?

First, this will probably be the last election where the legacy of the 1960’s and 1970’s will be fought out in this way on the left.  Barack Obama is, in many ways, an end-run around this conflict.  Follower of the 1960’s ideal (and radicals like Bill Ayers) he is; typical product of this country he is not.  But now the day of reckoning has come.

Second, the left isn’t as unified as its media boosters would have us believe.  If the Democrats ever get to a permanent majority, the first result of it will be a split, pretty much along the divide we’re seeing now.  The non-white constituencies simply do not operate in the same way as their (mostly) upscale white ones do; we’re already seeing that in the re-segregation of our college campuses and “safe spaces” for different groups.  The real end-game for identity politics is a type of millet system, but the Democrats are simply too “American” to take full advantage of this, at least on a national scale.

The Democrats have this last problem, but the Republicans have it even more.  The Republican Party has proven its ability to produce a diverse field for President, but its own ideological commitments will make outreach to those groups difficult.

The question neither side has a really good answer for is whether their idea will make for a greater country.  In the case of the hippie dreamers on the left, such a question is absurd: to their mind the world will be a better place if America is diminished, and Obama’s foreign policy is a reflection of that conviction.  On the right translating populism into success is a tricky proposition, which is one reason Donald Trump’s candidacy is so problematic.

In the next few weeks we’ll see which hippie dreamer comes out on top.  This election is very important, but it’s really hard to see a good outcome.  At best we can hope for yet another holding pattern while avoiding being caught in the crossfire between the hippie dreamers.

When Public Servants Were Barred from Being Priests

Such was the case in the late Roman Empire, by no less of a personage than Pope Siricius.  Writing to the bishops in North Africa, he gave eight reasons why a person should not be consecrated to the priesthood, saying that “if after the remission of sins (baptism) he (the candidate for the priesthood) takes on the belt of public service it is not right for him to be admitted to clerical orders.” (Letter V, 1) The “belt of public service” was part of the uniform of the Roman bureaucracy, which took on military form in its civilian part as well.  The Council of Toledo made the same prohibition in 401.  Siricius expressed the same disapproval of people passing from civil service to the priesthood in a later letter to several bishops (Letter VI, I, 3.)

Why was this? Siricius and others were well aware of the nature of late Roman politics, which involved patronage and graft, to say nothing of torture and execution.  He could not imagine someone successful in the Roman bureaucracy having the moral character necessary to be a Christian priest or bishop.  For all the trashing of fourth century Christianity by some of those who came after, this is a higher standard than much of what we see these days.  We are better at making our own system look clean, but there is plenty of corruption to go around.

And when the opportunity to unload this bureaucratic weight came around, as Britain did a few years later, the glee was clear, as we can see in the Pelagian Fastidius’ De Vita Christiana:

We see before us plenty of examples of wicked men, the sum of their sins complete, who are at this present moment being judged, and denied this present life no less than the life to come…Those who have freely shed the blood of others are now forced to spill their own…Some lie unburied, food for the beasts and birds of the air…Their judgements killed many husbands, widowed many women, orphaned many children.  They made beggars and laid them bare…for they plundered the children of the men they killed.  Now it is their wives who are widows, their sons who are orphans, begging their daily bread from others.

Today, of course, Christians are made to think that participation in public life is their Christian duty, but there was a time when just the opposite was commended to Christ’s followers.  In both cases good reason is involved; it is not as easy an issue as some think.

A Letter from the Rector

I was looking through some papers and found a letter from an Episcopal rector with this:

I did enjoy your letter and it just makes me that much more distressed that you left the Episcopal Church. Somehow, with your mind and keen feelings, we should have been able to hang on to you. We sorely need the prayers of everyone and their understanding during this time of crisis in the Church. It would be so easy to “throw in the sponge” and go along with the crowd, but my disposition is not such.  I suppose I will go down fighting for what I feel I have to do.

Personally, I do not think there is much hope for the Episcopal Church at the present time except to grow smaller and smaller as more and more people leave it to go elsewhere, or to join with the Anglican body now being formed.

And the date? Perhaps in the last decade or so, after the crisis detonated by Vickie Gene Robinson’s elevation to bishop? Hardly.  The letter was written in January 1978, the rector was the Very Rev. James C. Stoutsenberger, and the parish was St. Joseph’s Episcopal Church in Boynton Beach, FL.

Before I get to commenting on this “contemporary feeling” epistle, some background is in order.

My home church is Bethesda-by-the-Sea in Palm Beach; however, in 1972 we moved to Boynton Beach.  I was the only one in my family going to church anywhere at that point, and that was St. Thomas More Catholic Church, the destination for my “Tiber swim.” A few years later church attendance became a “political football” in my parent’s protracted divorce, and that’s where Rev. Stoutsenberger came in.

Now for some observations about this letter, which could have been written a quarter century after it was:

  1. The “Anglican body” he was referring to was that of the predominantly Anglo-Catholic “continuing” Anglicans, which had met and issued a statement the prior year.  As we all know, they formed a few parishes and dioceses, but really didn’t make much of a dent in TEC.
  2. At the time most of the impetus to form a new body came from the Anglo-Catholics.  The Charismatics, like their LGBT counterparts, were too fixated on changing existing institutions and not making new ones. The Reformed, I suppose, were simply out to lunch in those days, or “swimming the Tweed.”
  3. It’s interesting to think what would have happened if these continuing Anglicans had really taken hold at the time; the Dennis Canon wasn’t passed (or was it?) until the following year.  It would have definitely levelled the playing field had those seceding not had to deal with it.
  4. The Episcopal Church’s lurch left and the membership bleed that followed isn’t a recent phenomenon; it was just Round II, Round I having taken place in the 1960’s and 1970’s.  “Smaller and smaller” has been the trajectory of TEC ever since, except they managed to stop the bleeding in the 1980’s and 1990’s long enough to gather people in who weren’t there for the first drop, but many of whom were involved in the second.
  5. Most of the people who left at the time and stayed in Christianity either swam the Tiber like I did or went to an Evangelical or Charismatic church of some kind.
  6. I think one major reason the continuing Anglicans didn’t make the impact their later, AMiA/ACNA/CANA counterparts did was the lack of a ready means to make a community and spread the message.  The internet handed the Anglican world just that in the 1990’s, and the rest is, as they say, history.
  7. Another reason was the continuing churches’ lack of communion with Canterbury, an obsession which has lurked in the Anglican/Episcopal psyche from the start. The AMiA, formed by the provinces of Rwanda and South-East Asia, fixed that problem to some extent, and now we have the results of the recent Primates’ meeting.

Stoutsenberger put “his money where his mouth was” and ended up serving at a FiFNA affiliated parish in Lantana. He passed away in 2004, living long enough to see the explosion that has brought us where we are.  It’s sad that it took the elevation of an openly gay man to motivate people, because TEC’s problems were clear long before that event.  That’s indicative of peoples’ low consciousness and understanding of what Christianity is all about, and that situation (in this country at least) shows little sign of improving.

Hillary’s Hidden Obstacle in the Electorate

Voting at eighteen years of age has been the law in the U.S. for nearly a half century now.  But for those who are off to college, the same age bracket starts voting in other ways, and one of those are student evaluations of professors.  There is a great deal of argument about how much stock to put in these.  My problem with them has always been that a professor that comes across great during the semester/term may not look too hot a few years in the rear view mirror, and vice versa.  That was certainly the case with mine.

In any case, this tidbit from two studies on the subject is startling:

Students’ gender appeared to impact their bias, but in different ways in the French and U.S. samples.

In the French data, male students tended to rate male instructors higher than they rated female instructors, but little difference was observed among female students. In the U.S. data, female students tended to rate perceived male instructors higher than they rated perceived female instructors, with little difference in ratings by male students. In both cases, however, the bias still positively impacted male instructors and disadvantaged female ones.

Too much extrapolation is always a danger, but if American female college students can’t bring themselves to bump up their faculty counterparts, how are they going to bring themselves to vote en bloc for an American woman for President?  Laura Schlesinger made an offhand comment one time that it wasn’t the men who wouldn’t vote for a woman for president, it was the women.  She may be on to something.

Back on campus, this leads to two interesting inferences from the data.

The first is that women who teach in largely male dominated majors (such as engineering) will come out better with their students than those where women predominate (such as education and the liberal arts.)  That doesn’t always translate into promotion for women by the administration, as the end of “Turkish rule” here at UTC’s College of Engineering and Computer Science will attest.

The second is that male professors have a better shot at student popularity in fields where women predominate.  This is a collegiate version of the “rooster phenomenon” which Pentecostals will recall with disdain.  Just because you send people to institutions of higher learning doesn’t mean that they will fundamentally change.

And neither will they change so much when they enter the voting booth, which is one more obstacle Hillary Clinton will have to overcome shortly.

Some Thoughts on Bossuet’s History of the Variations of the Protestant Churches

One of the things that some of the major Anglican blogs will throw out from time to time is the question of what their readers/commenters are reading on the side when they’re not keeping up with the latest Anglican debacle (like the recent Primates’ Meeting.)  Through the Christmas holidays, while waiting for some long runs to come out of the computer, I finished Bossuet’s History of the Variations of the Protestant Churches, both volumes of same.  That may seem quaint to some, although this piece on the recent Primates’ Meeting strikes me as being taken out of the Variations without Bossuet’s ability to entertain and inspire.  (Most Catholics priests these days lack Bossuet’s ability both ways, but that’s another post…)

The Variations were Bossuet’s efforts to show the serious problems inherent in the Reformed churches.  So how successful was he? Part of how successful he seems depends upon how you accept his view of Roman Catholicism.  A Roman Catholicism which is more like Bossuet envisions it–conscious of Scripture, independent of the state, Augustinian in theology–would be a better entity to adhere to than the one that he had than and we have now.  A big part of the problem is that the reverends pères jesuites, or at least one in particular (Pope Francis,) are once again propagating their morale accommodante, as they did in Bossuet’s France (much to its long-term detriment.)  Unfortunately then and now the situation is more complicated, but Bossuet tends to ignore this.

His invective against Protestantism, however, works, and it does because he picks his battles carefully.  Although it’s easy to get lost in his nit-picking of the endless declarations of faith (they contradicted each other and Catholic doctrine,) the largest thing he goes after is the complete hash that Protestant churches made over the nature of the Eucharist.  It was the first major split in Protestantism, pitting Lutheran consubstantiation (with its multiple definitions) against Bill Clinton’s Eucharistic Theology as advocated by Zwingli, and the variations which followed…Bossuet succeeds in showing that, once you get away from the literal meaning of Christ’s words instituting the Eucharist, you get a mess compared to which the problems of transubstantiation pale.

That’s not the only thing Bossuet occupies his pages with, though.  Except for the Anabaptists and what we would call the “Radical Reformation,” he covers his subject pretty well.  Needless to say, his point of view is biased.  It becomes ferociously so when he gets to the English Reformation, which he depicts as a combination of duplicity and brutal state coercion (he conveniently ignores Queen Mary, but she simply kept up the pace set by her father, only for the other side.)  The one person that comes out of the narrative with her reputation intact is Queen Elizabeth I, whose settlement pulled back from the outlying positions of the Reformers (much to the distaste of the Puritans and other dissenters who spent the next century trying to pull in the other direction.)  For people who are enamoured with the myth-making of the English-speaking peoples, Bossuet’s viewpoint is hard to swallow but necessary.

Another interesting digression of Bossuet’s was his narrative of pre-Reformation groups such as the Albigensians, Waldensians (the “Vaudois” as he calls them) and the Bohemian Brethren.  The Reformers saw them as their forerunners and, indeed, looked to groups such as this as proof that there was always a “true church.” (This last point was revived in the nineteenth century in the “Baptist Succession” idea of J.R. Graves and those who came after him.)  Bossuet shows that the theology of these groups was at serious variance with what the Reformers taught, which led the latter to try to bring the former into line with their own idea.  Especially interesting are the Vaudois, who were in reality an unauthorised, non-celibate religious community in Catholicism more than a stand-alone church; their main fault is that they believed that unworthy priests did not administer valid sacraments.  (Anyone who has been in church work knows that gauging the worthiness of ministers can be a dicey proposition at best; I think the Vaudois were unreasonable in that regard.)

To my mind, the best part of the work was when Bossuet takes on Calvinism.  He hits the nail on the head when he characterises it as follows:

This doctrine of Beza was taken from Calvin, who maintains, in express terms, “that Adam could not avoid falling, yet was nevertheless guilty, because he fell voluntarily;” which he undertakes to prove in his Institution, and reduces the whole of his doctrine to two principles: the first, that the will of God causes in all things, even in our wills, without excepting that of Adam, an inevitable necessity; the second, that this necessity is no excuse for sinners.  Hereby it is plain, he preserves free will in name only, even in the state of innocence and after this there is no room for disputing whether he makes God the author of sin, since besides his frequently drawing this consequence, it is but too evident, by the principles he lays down, that the will of God is the sole cause of that necessity imposed on all that sin.

Bossuet goes on to show two characteristics of Reformed types that persist to this day: they spend half the time in their unbending insistence of their idea, and the rest of it back-pedalling from the fatalistic consequences of that idea.  The first was certainly in evidence in the smack-down that the Arminians experienced at the Synod of Dort, and the second started afterwards.  Much of the later history of Protestantism–especially the Wesleyan movement and its progeny–has been trying to fix this serious doctrinal problem, but given the Reformed strength in both the seminaries and the upper socio-economic strata of Christianity, it will always be an uphill battle.

As I alluded to earlier, Bossuet is an Augustinian; nevertheless, he has no sympathy with those who wanted to take Augustinianism (especially Calvin) in a new direction.  He also lays to rest Chesterton’s charge that Luther, an Augustinian monk himself, took Augustine’s doctrine (which certainly has problems of its own) to its logical conclusion.  Bossuet’s case for his own church would have been stronger had the Jesuits (with the backing of his own sovereign, Louis XIV) had not been undermining it with their casuistry, which Pascal (someone Bossuet was certainly familiar with) attacked with gusto in the Provincial Letters.

No matter where you’re at on the issues Bossuet discusses, the History of the Variations of the Protestant Churches is an interesting take on the Reformation, a process which did not end with Luther, Calvin and Cranmer but only began.

My Thoughts on the Anglican Primates’ Meeting

It’s just about over, and the Primates meeting in Canterbury have made their official statement, such as it is.  Here are some observations:

  1. I’ve always felt that it was unrealistic to expect the current Archbishop of Canterbury to allow TEC and ACoC to get the boot.  In that context what happened, i.e., TEC being put in the “penalty box” for three years, was more than I expected, especially in view of the manipulative way Welby handled the meeting.
  2. I was also surprised that the gathering “upholds marriage as between a man and a woman in faithful, lifelong union.”  Obviously most of the primates gathered believe that; getting an Anglican group to be that plain about saying it is another matter altogether.  Evidently the GAFCON primates, along with a growing group from some of the other provinces, are having some impact.
  3. Officially, ACNA got nothing out of this.  I am sure that Canterbury aficionados in the ACNA felt the thrill up their leg at Archbishop Foley’s presence at this meeting, but it really doesn’t amount to much.  Welby has always tipped his hat to ACNA without really giving them what they’re looking for, i.e., designation as the “official” Anglican church in the U.S., the TEC itself getting the boot.
  4. Evidently TEC Presiding Bishop Curry is reverting to good old Episcopal “mealymouth” in taking the Primates’ decision in stride.  He knows that GC will not back down on same-sex marriages, so nothing will change in TEC.  Given TEC’s current membership erosion and financial woes, Curry may have been handed a nice excuse to cut back on their contribution to the Anglican Communion Office; he’s got more pressing problems right at the moment.
  5. The omission of the Canadians in the penalty box can only be described as bizarre.
  6. I still think that Welby is in a tight place at home with officiating same-sex marriages; sooner or later the CoE will be forced to, if it does not capitulate in advance.  That would be a game changer that the ACNA, and to a lesser extent GAFCON, aren’t quite ready to effectively deal with.

All this being the case, I still think that the GAFCON provinces and their allies need to make other arrangements.  What happened this week only drags things out.  The Anglican Communion, like Brunei, is a good place to watch the grass grow.

The Places I Couldn’t Teach

The flap over Wheaton’s process to dismiss Larycia Hawkins from her position makes me stop and think about a few things, especially since I am beginning yet another semester of teaching Civil Engineering at UTC.  Lord willing, sometime this year I will complete my PhD pursuit.  It’s been a long process, not without excitement; hopefully I’ll be able to put things in perspective as it comes to an end.

One of the things that I’ve actually accomplished along the way is to accumulate enough graduate hours to teach math at an accredited institution (well, SACS at least.)  By some accounts, it’s not the highest and best use for this PhD but it’s possible.  Most Christian colleges offer math courses, although if one considers some of the things our ministers say it’s easy to conclude that whatever math they had to take didn’t make much impact.

You’d think that math would be a relatively uncontroversial subject, without much of the doctrinal baggage that would cloud a liberal arts professor, to say nothing of those who teach “divinity.”  But that’s not necessarily the case.  Christian colleges, in trying to be consistent, usually require all faculty to adhere to a doctrinal statement, and that include those who count.

Let’s consider, for example, Bryan College, just up the road in Dayton.  It’s a nice place, my wife and I have visited for a number of cultural events, know President Stephen Livesay and his wife, have friends on the faculty.  But I could never teach there because I’m a shameless old earther, and Bryan requires that a faculty member be a six-day young earther.  It is the same at Patrick Henry.  Wheaton’s situation is a little dodgier, but since they consciously exclude Roman Catholics, would I be out of luck since I believe in the real presence of Christ in the Eucharist?

At this point I need to make a couple of stipulations.

The first is that I promised myself a long time ago that my work would be a “diverse portfolio” to avoid being the client of a single patron.  My years of working for the church only underscore that commitment.

The second is that I don’t think that the American concept of “accommodation” is a New Testament requirement of institutions, Christian or otherwise.  The idea is really recent, i.e., the Civil Rights Act of 1964.  This has led, IMHO, to a lot of the “we can have it all” attitude that too many Evangelicals are stuck on, that we can do anything we want and it still be compatible with our faith.

Those things said, I think it is the prerogative of institutions such as Bryan and Wheaton to make the doctrinal requirements that they do and take the actions they deem appropriate to actualise those requirements.  However, they must also rise and fall on the consequences of those requirements, be they good or bad.  Ultimately their opponents should support the institutions that support their view, or start ones that do if they must, instead of expecting institutions to constantly bend to their will.

The last point has been the tricky part for the religious left.  When UTC Provost Jerald Ainsworth introduced Dr. Daniel Pack, the new Dean for UTC’s College of Engineering and Computer Science, he said that what he was looking for was not a maintainer or a fixer but a builder.  The left has been good at taking control of institutions–secular and sacred–but not so hot at either building them up or making new ones to supplant those not to their taste.  As long as that is true, Christian institutions will be the arena for slugfests like the one we’re seeing at Wheaton.

In the meanwhile, it’s time for the rest of us to roll on.

Evangelicals Having “Buyers Remorse” on Being Pro-Life?

Sure looks that way, at least for the organisers of Urbana15:

In an op-ed published on Monday, Kristan Hawkins, president of Students for Life (SFL), revealed the Urbana15 team denied her group’s exhibitor application.

SFL received an email from Urbana’s Exhibits Manager thanking the pro-life youth organization for applying, but denied their application because, “… Students for Life does not align with Urbana’s exhibitor criteria. One of our key criteria for exhibitors is to have advancing God’s global mission as the vision and purpose of their organization.”

It’s easy to forget, but at the time of Roe v. Wade, evangelicals were decidedly unenthusiastic about the pro-life cause.  With Roman Catholics it was another story, although to some extent that was muted in the upheaval following Vatican II.  Evangelicals generally took a blasé attitude towards the subject.  It was more important, to their mind, to work on evangelising those who made it to the age of accountability rather than to fret over those who didn’t, as they had no worries about their eternal destiny.

It took some promotion, but by the 1980’s Evangelicals and the “Religious Right” were in the forefront of the pro-life movement, to the point where there are people out there who think that the Roman Catholic obsession with the subject came from the Evangelicals!

Today, for conservative Roman Catholics, pro-life is the social issue, even taking precedence over same-sex marriage.  And there are Evangelicals for whom it is the same, as is probably the case with most of the Students for Life.  However, in the ever-running popularity contest of Evangelicalism, some have decided that the pro-life cause carries too much baggage, and thus it gets banned from an evangelistic gathering like Urbana15.  It’s like, after forty years or so of making the pro-life movement central, Evangelicalism is showing signs of “buyers remorse” for a cause they didn’t much care for to start with.

Personally I think both getting people into the world and getting them saved after that are important.  But the Body of Christ is supposed to be equipped with diverse gifts and callings, right? So do we all really have to do the same thing? Evidently in this age of enforced groupthink this is too much for some Christian leaders.

If being pro-life is the thing for you, you’re probably better off being Roman Catholic than Evangelical.

Consistency, Convergence and Stability of Lax-Wendroff Scheme Applied to Convection Equation

The purpose of this project is to examine the Lax-Wendroff scheme to solve the convection (or one-way wave) equation and to determine its consistency, convergence
and stability.

Overview of Taylor Series Expansions

Both of the cases examined utilize Taylor Series expansions, so some explanation common to both is in order. The general expression for a Taylor series is found in A Course in Mathematical Analysis Volume 1: Derivatives and Differentials; Definite Integrals; Expansion in Series; Applications to Geometry (Dover Books on Mathematics) and is given as

Taylor-Series

As a general rule, h will represent a time or distance step, i.e. \Delta_{x},\,\Delta_{t}, although the second case will require a more versatile application of h.

In any event, the forward spatial Taylor series expansion from a single point is given as

u(x_{{k+1}},t_{{n}})=u(x_{{k}},t_{{n}})+D_{{1}}(u)(x_{{k}},t_{{n}})\Delta_{{x}}+1/2\,\left(D_{{1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{2}+1/6\,\left(D_{{1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{3}\\ +1/24\,\left(D_{{1,1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{4}+{\frac{1}{120}}\,\left(D_{{1,1,1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{5}\\+{\frac{1}{720}}\,\left(D_{{1,1,1,1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{6}+O(1){\Delta_{{x}}}^{7}

For our analysis u\left(x,t\right) is the function of the finite difference approximation, contrasted with the exact function v\left(x,t\right). The subscripts k,n and indices for space and time respectively. The backward spatial expansion is given as

u(x_{{k-1}},t_{{n}})=u(x_{{k}},t_{{n}})-D_{{1}}(u)(x_{{k}},t_{{n}})\Delta_{{x}}+1/2\,\left(D_{{1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{2}\\-1/6\,\left(D_{{1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{3}+1/24\,\left(D_{{1,1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{4}\\-{\frac{1}{120}}\,\left(D_{{1,1,1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{5}+{\frac{1}{720}}\,\left(D_{{1,1,1,1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{6}-O(1){\Delta_{{x}}}^{7}

In like fashion the expansion for time is as follows:

u(x_{{k}},t_{{n+1}})u(x_{{k}},t_{{n}})+D_{{2}}(u)(x_{{k}},t_{{n}})\Delta_{{t}}+1/2\,\left(D_{{2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{2}\\+1/6\,\left(D_{{2,2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{3}+1/24\,\left(D_{{2,2,2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{4}\\+{\frac{1}{120}}\,\left(D_{{2,2,2,2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{5}+{\frac{1}{720}}\,\left(D_{{2,2,2,2,2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{6}+O(1){\Delta_{{t}}}^{7}

Convection Equation

Now let us turn to the convection equation.  Although CFD aficionados refer to this equation in this way, in solid mechanics this is the “one-way” wave equation, i.e., without reflections. The derivation and solution of this equation is detailed here.

In either case the governing equation is

{\frac{\partial}{\partial t}}v(x,t)+a{\frac{\partial}{\partial x}}v(x,t)=0

When solved using the Lax-Wendroff scheme, it is expressed as

u(x_{{k}},t_{{n+1}})=u(x_{{k}},t_{{n}})-1/2\, R\left(u(x_{{k+1}},t_{{n}})-u(x_{{k-1}},t_{{n}})\right)\\+1/2\,{R}^{2}\left(u(x_{{k+1}},t_{{n}})-2\, u(x_{{k}},t_{{n}})+u(x_{{k-1}},t_{{n}})\right)

where

R=a\frac{\Delta_{t}}{\Delta_{x}}

The solution for this problem is given in Numerical Methods for Engineers and Scientists, Second Edition.

Application of Taylor Series Expansions for Consistency

If we apply the results of the Taylor series expansions to the Lax-Wendroff scheme and perform a good deal of algebra (including substituting for R,) the result is

u(x_{{k}},t_{{n}})+D_{{2}}(u)(x_{{k}},t_{{n}})\Delta_{{t}}+1/2\,\left(D_{{2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{2}+1/6\,\left(D_{{2,2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{3}\\+1/24\,\left(D_{{2,2,2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{4}+{\frac{1}{120}}\,\left(D_{{2,2,2,2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{5}+{\frac{1}{720}}\,\left(D_{{2,2,2,2,2,2}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{t}}}^{6}+O(1){\Delta_{{t}}}^{7}\\=\\u(x_{{k}},t_{{n}})+r\left(D_{{1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{2}-1/12\, r\left(D_{{1,1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{4}-1/40\, r\left(D_{{1,1,1,1,1,1}}\right)(u)(x_{{k}},t_{{n}}){\Delta_{{x}}}^{6}

Rearranging, making a change in notation and dropping the \mathcal{O} terms as well,

Convection-Equation-4

On the left hand side is the exact equation, which is strictly speaking equal to zero. On the right hand side is the residual for consistency of the finite difference scheme. If the scheme is consistent with the original equation, it too should approach zero as \Delta_{x},\,\Delta_{t}\rightarrow0. Based on this we note the following:

  • All of the right hand side terms contain \Delta_{x},\,\Delta_{t} or both. Thus, as these approach zero, the entire right hand side will approach zero. Thus the scheme is consistent with the original differential equation.
  • The lowest order terms for the time and spatial steps on the right hand side are \Delta_{t} and \Delta_{x}^{2} respectively. Thus we can conclude that the truncation error is \mathcal{O}\left(t\right)+\mathcal{O}\left(x^{2}\right).

Or is it? Let us assume that the solution is twice differentiable. By this we mean that the function has second derivatives in both space and time. (Another way of interpreting this is to say that “twice differentiable” means that the solution has no derivatives beyond the second, in which case many of the terms in the Taylor Series expansion would go to zero.) Then we differentiate the original equation once temporally, thus

{\frac{\partial^{2}}{\partial{t}^{2}}}v(x,t)+a{\frac{\partial^{2}}{\partial t\partial x}}v(x,t)=0

Now let us do the same thing but spatially, and (with a little additional algebra) we obtain

-a{\frac{\partial^{2}}{\partial t\partial x}}v(x,t)-{a}^{2}{\frac{\partial^{2}}{\partial{x}^{2}}}v(x,t)=0

Adding these two, we have

{\frac{\partial^{2}}{\partial{t}^{2}}}v(x,t)-{a}^{2}{\frac{\partial^{2}}{\partial{x}^{2}}}v(x,t)=0

which is, mirabile visu, the wave equation. Applying this solution for the original equation to the finite difference residual results in

Convection-Equation-6Now we see that the lowest order terms are \Delta_{t}^{2} and \Delta_{x}^{2},  which means that the truncation error is \mathcal{O}\left(t^{2}\right)+\mathcal{O}\left(x^{2}\right). We duly note that the fourth order spatial derivative is multiplied by \Delta_{{t}}{\Delta_{{x}}}^{2}. However, the squared term will be the predominant one as \Delta_{x},\,\Delta_{t}\rightarrow0, so this does not change our conclusion. Also, if “twice differentiable” means that the function has no further derivatives beyond the second, then all of the terms go to zero, and the numerical solution, within machine accuracy, is exact. This also applies to the next section as well; the vector described there would be the zero vector under these conditions.

Consistency in a Norm

The Taylor Series expansion is only valid at the point at which it is taken. For most differential equations, we are interested in solutions along a broader region. This is in part the purpose for considering consistency in a norm.

Let us consider the result we just obtained, thus

Convection-Equation-7aThe right hand side represents the residual for consistency of the finite difference scheme. If we were to consider a Taylor Series expansion for all of the points in space and time under consideration, what we would end up with is an infinite set of residuals, i.e., the right hand side of the above equation, which could then be arranged in a vector. If we designate this vector asR, then each entry can be
designated as follows:

Convection-Equation-7b
Now let us consider the nature of the differential equation. The following is adapted from Numerical Solution of Differential Equations: Finite Difference and Finite Element Solution of the Initial, Boundary and Eigenvalue Problem in the Ma (Computer Science and Applied Mathematics)
.

We can consider the differential equation as a linear transformation. Since we have defined the results as an vector, we can express this as follows for the exact solution:

Av\left(x,t\right)=F

and for the finite difference solution

Au\left(x,t\right)=F+R

The result difference between the two is the residual we defined earlier. The finite difference representation is the same as the original if and only if A is the same in both cases. Combining both equations,

A\left(v\left(x,t\right)-u\left(x,t\right)\right)=R

and rearranging

v\left(x,t\right)-u\left(x,t\right)=A^{-1}R

Now let us consider the norm. Given the infinite number of entries in this vector, the most convenient norm to take would be the infinity norm, where the norm would be the largest absolute value in the set. For an inner product space,

||v\left(x,t\right)-u\left(x,t\right)||_{\infty}=||A^{-1}||_{\infty}\,||R||_{\infty}

We have shown that each and every r_{n}\rightarrow0 as \Delta_{x},\,\Delta_{t}\rightarrow0. (Additionally the function would have to be bounded, continuous and at least twice differentiable at all points.) From this, R\rightarrow0 and ||R||_{\infty}\rightarrow0. If A and A^{-1} are bounded (as they are in a linear transformation,) then ||v\left(x,t\right)-u\left(x,t\right)||_{\infty}\rightarrow0 and thus the exact solution and its finite difference counterpart become the same. This is consistency by definition. (The most serious obstacle to actually constructing such a vector–a necessary prerequisite to a norm–is evaluating the derivatives. One “solution” would be to used the exact solution of the original differential equation, but that assumes we can arrive at an exact solution. In many cases, the whole point of a numerical solution is because the exact, “closed form” solution is unavailable. Thus we would end up with numerical evaluations for the derivatives.)

As for other norms such as the Euclidean norm, if the entries in the vector approach zero as \Delta_{x},\,\Delta_{t}\rightarrow0, we would expect the norm to do so as well, as discussed above. It should be noted that the infinity norm would be best to pick up a point slowly converging on zero than a Euclidean norm.

von Neumann Stability Analysis

Turning to the issue of stability, we will perform a von Neumann analysis. In this type of analysis we will analyze a stability factor |G| defined as follows:

1\geq|G|=\frac{u(x_{{k}},t_{{n+1}})}{u(x_{{k}},t_{{n}})}

The idea behind this is to determine “whether or not the calculation can be rendered useless by unfavourable error propagation” (from The Numerical Treatment of Differential Equations.) In other methods, such as perturbation methods, an error is introduced into the scheme and its propagation is explicitly analysed. The von Neumann analysis does the same thing but in a more compact form.

The heart of the von Neumann method is to substitute a Fourier series expression into the difference scheme. Thus, for our difference scheme

u(x_{{k}},t_{{n+1}})=u(x_{{k}},t_{{n}})-1/2\, R\left(u(x_{{k+1}},t_{{n}})-u(x_{{k-1}},t_{{n}})\right)+1/2\,{R}^{2}\left(u(x_{{k+1}},t_{{n}})-2\, u(x_{{k}},t_{{n}})+u(x_{{k-1}},t_{{n}})\right)

we substitute
u(x_{{k}},t_{{n+1}})={e^{p_{{m}}\left(t+\Delta_{{t}}\right)+\sqrt{-1}k_{{m}}x}}\\u(x_{{k}},t_{{n}})={e^{p_{{m}}t+\sqrt{-1}k_{{m}}x}}\\u(x_{{k+1}},t_{{n}})={e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x+\Delta_{{x}}\right)}}\\u(x_{{k-1}},t_{{n}})= {e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x-\Delta_{{x}}\right)}}

to yield

{e^{p_{{m}}\left(t+\Delta_{{t}}\right)+\sqrt{-1}k_{{m}}x}}={e^{p_{{m}}t+\sqrt{-1}k_{{m}}x}}\\-1/2\, R\left({e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x+\Delta_{{x}}\right)}}-{e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x-\Delta_{{x}}\right)}}\right)\\+1/2\,{R}^{2}\left({e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x+\Delta_{{x}}\right)}}-2\,{e^{p_{{m}}t+\sqrt{-1}k_{{m}}x}}+{e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x-\Delta_{{x}}\right)}}\right)

As an aside, When most people think of “Fourier series” they think of a real series of sines, cosines and coefficients. This was certainly in evidence in the presentation of the method in The Numerical Treatment of Differential Equations. However, it has been the author’s experience that the best way to treat these is to do so in a “real-complex continuum,” i.e., to express these exponentially and to convert them to circular (or in some cases hyperbolic) functions as the complex analysis would admit. An example of that is here.

Solving for the amplification factor defined above,

|G|={\frac{{e^{p_{{m}}t+\sqrt{-1}k_{{m}}x}}-1/2\, R\left({e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x+\Delta_{{x}}\right)}}-{e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x-\Delta_{{x}}\right)}}\right)}{{e^{p_{{m}}t+\sqrt{-1}k_{{m}}x}}}}\\+{\frac{1/2\,{R}^{2}\left({e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x+\Delta_{{x}}\right)}}-2\,{e^{p_{{m}}t+\sqrt{-1}k_{{m}}x}}+{e^{p_{{m}}t+\sqrt{-1}k_{{m}}\left(x-\Delta_{{x}}\right)}}\right)}{{e^{p_{{m}}t+\sqrt{-1}k_{{m}}x}}}}

Simplifying,
|G|=1-1/2\, R{e^{\sqrt{-1}k_{{m}}\Delta_{{x}}}}+1/2\, R{e^{-\sqrt{-1}k_{{m}}\Delta_{{x}}}}+1/2\,{R}^{2}{e^{\sqrt{-1}k_{{m}}\Delta_{{x}}}}-{R}^{2}+1/2\,{R}^{2}{e^{-\sqrt{-1}k_{{m}}\Delta_{{x}}}}

or
|G|=1+{R}^{2}\cos(k_{{m}}\Delta_{{x}})-{R}^{2}-\sqrt{-1}R\sin(k_{{m}}\Delta_{{x}})

and then

1\geq G=\sqrt{\left(1+{R}^{2}\cos(k_{{m}}\Delta_{{x}})-{R}^{2}\right)^{2}+{R}^{2}\left(\sin(k_{{m}}\Delta_{{x}})\right)^{2}}

Solving for R at the points of equality yields three results: R=-1,0,1. Since negative values for R have no physical meaning, we conclude that

0\leq R\leq1

for this method to be stable. Thus we can say that the method is conditionally unstable.

Convergence

The Lax Equivalence Theorem posits that, if the problem is properly posed and the finite difference scheme used is consistent and stable, the necessary and consistent conditions for convergence have been met (see Numerical Methods for Engineers and Scientists, Second Edition.) We have shown that, with the assumptions stated above, the scheme is consistent with the original differential equation,
with or without the provision of twice differentiability. The method is thus convergent within the conditions stated above for stability; outside of those conditions the method is neither stable nor convergent.

Sailing the Last Voyage with Newton and Pascal

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