Symbolic Generation of Finite Difference Formulae Robert M. Corless Department of Applied Mathematics and the Ontario Research Centre for Computer Algebra University of Western Ontario This work is joint work with Jichao Zhao, a Ph.D. student working under my direction, and is based in part on work done with Jacek Rokicki (T.U.Warsaw). The applications in mathematical finance (not talked about here) are joint work with Matt Davison (Applied Math, Western) and Jichao Zhao.

The finite difference calculus is very old, going back at least to Newton; even as a topic in modern numerical analysis it is very old. It would be surprising if there were something really new to say...today we will have a tour around some old results, and do some bookkeeping & programming, and make a neat, though minor, refinement on a classical but recently more popular finite difference technique, now called "compact finite differences". We will end (time permitting) with a discussion of some stability aspects. From "A Treatise on the Calculus of Finite Differences" by George Boole, 1st ed. 1860, 2nd ed. 1872 edited by J. F. Moulton: "The Calculus of Finite Differences may be strictly defined as the science which is occupied about the ratios of the simultaneous increments of quantities mutually dependent." From "The Calculus of Finite Differences" by L. M. Milne-Thompson, 1933: "Let f(x) be a given function of the variable x. The Differential Calculus is concerned with the properties of 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 which is still a function of the single 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which is a function of the two variables 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 and 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." Familiar finite differences include the forward difference above, which approximates the derivative to order h: Delta[h](f)(x) = (f(x+h)-f(x))/h;

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

lhs() = series(rhs(),h,3);

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

The central difference operator which approximates the derivative to order h^2: delta[h](f)(x) = (f(x+h/2)-f(x-h/2))/h;

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lhs() = series(rhs(),h,4);

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The old-fashioned usages of finite difference formulae were different from today's needs (indeed the first use of the Method of Lines was to derive a partial differential equation, not solve it! (see Hairer, Norsett & Wanner, Solving ODE I, section I.6, for a discussion of the contributions of Brook Taylor 1715, Johann Bernoulli 1727, and d'Alembert 1747), and by and large their operational methods for generation of formulae have fallen by the wayside: we know pretty much all the formulae we need. Or do we?

Generation of finite difference formulae is, nowadays, pretty easy: just look them up, many of them are known already. But, how can we find them ourselves?

The older texts make the connection between operator expansions and series: restart; f(x+h) = series( f(x+h),h );

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The following is the "shift" operator; it takes a function, and returns a function with shifted argument: E := f -> (t->f(t+h));

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

E(sin);

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

E(sin)(x);

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

Composition of operators is represented by the symbol @. Repeated composition is by @@, analogous to the old Fortran * for times and ** for power. (E@@3)(sin)(x);

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"It is obvious that" Taylor expansion means that the operators E and D may be related by the following series, which looks like the exponential series: E = I + h*D + h^2*(D@@2)/2! + h^3*(D@@3)/3! + O(h^4);

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In other words, for a slightly modified meaning of the exponential function, E = exp(h*D);

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Of course, that statement is a shorthand notation for an operator relation: Delta = E - I;

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Delta = (exp(h*D) - 1)/h;

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

Dln := solve(%,D);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USREbG5GKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYoLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GKC8lJ29wYXF1ZUdGOC8lK2V4ZWN1dGFibGVHRjgvJSlyZWFkb25seUdGOC8lKWNvbXBvc2VkR0Y4LyUqY29udmVydGVkR0Y4LyUraW1zZWxlY3RlZEdGOC8lLHBsYWNlaG9sZGVyR0Y4LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRy8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjUtSSNtb0dGJTYzUSM6PUYoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRigvJSdyc3BhY2VHRltwLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGWC8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRy1JJm1mcmFjR0YlNiotRiQ2Iy1GJDYlLUYtNjlRI2xuRihGMEYzRjYvRjpGOEY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ24vRmpuUSdub3JtYWxGKEZcby1GX282M1EwJkFwcGx5RnVuY3Rpb247RihGYm9GZW9GZ28vRmpvUSQwZW1GKC9GXXBGZ3JGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2JS1GX282M1EiKEYoL0Zjb1EncHJlZml4RigvRmZvRjtGZ28vRmpvUS50aGlubWF0aHNwYWNlRigvRl1wRmJzL0ZfcEY7RmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUYkNiMtRiQ2JS1GJDYlLUYtNjlRJkRlbHRhRihGMEYzRjZGYHJGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmFyRlxvLUZfbzYzUTEmSW52aXNpYmxlVGltZXM7RihGYm9GZW9GZ29GZnJGaHJGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRi02OVEiaEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUZfbzYzUSIrRihGYm9GZW9GZ28vRmpvUTBtZWRpdW1tYXRoc3BhY2VGKC9GXXBGaHRGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtSSNtbkdGJTY5RmdwRjBGM0Y2RmByRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZhckZcby1GX282M1EiKUYoL0Zjb1EocG9zdGZpeEYoRmBzRmdvRmFzL0ZdcFEydmVyeXRoaW5tYXRoc3BhY2VGKEZkc0ZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcS1GJDYjRmF0LyUubGluZXRoaWNrbmVzc0dRIjFGKC8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGKC8lKW51bWFsaWduR0Zbdi8lKWJldmVsbGVkR0Y4RmJxRmRxNyMtX0YpSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREbG5HRigqJi1JI2xuRzYkJSpwcm90ZWN0ZWRHRio2IywmKiZJJkRlbHRhR0YoIiIiSSJoR0YoRmF3RmF3RmF3RmF3RmF3RmJ3ISIiNyNGaHY=

Wait a minute, that only LOOKED like an exponential---now this one LOOKS LIKE a logarithm... series( %, h );

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

D = %;

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

Let's check this, that seems somewhat strange: D(f)(x) - (f(x+h)-f(x))/h + (f(x+2*h)-2*f(x+h)+f(x))/2/h ;

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

series(%,h,4);

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So this technique appears to work! Can we soup it up? (f(x+h/2) - f(x-h/2))/h;

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series( %, h, 8 );

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

Of course everyone immediately recognizes that series as being a hyperbolic sine... series( 2*sinh(h*D/2)/h, h, 8 );

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(exp(h*D/2) - exp(-h*D/2))/h = convert( (exp(h*D/2) - exp(-h*D/2))/h, trigh );

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

delta = 2*sinh( h*D/2 )/h;

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Ddelta := solve(%,D);

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

D = series(%,h,8);

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delta := f -> (t->(f(t+h/2)-f(t-h/2))/h);

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The following is a "Maple trick" to ensure that h is treated as a constant in the following applications of operators. h := t -> h;

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USJoRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHUSV0cnVlRigvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYoLyUqbWF0aGNvbG9yR0ZELyUvbWF0aGJhY2tncm91bmRHRkcvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y1LUkjbW9HRiU2M1EjOj1GKC8lJWZvcm1HUSZpbmZpeEYoLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYoLyUncnNwYWNlR0ZbcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlgvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkctRiQ2JS1GLTY5USJ0RihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tRl9vNjNRJyZyYXJyO0YoL0Zjb1EhRihGZW9GZ28vRmpvUSQwZW1GKC9GXXBGYXJGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHFGLDcjLV9GKUksbXByaW50c2xhc2hHRig2JDcjPkkiaEdGKGYqNiNJInRHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKEZqckYoRihGKDcjRltz

D6 := delta - h^2*(delta@@3)/24 + 3/640*h^4*(delta@@5);

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

expand(D6(f)(t));

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

series(%,h,8);

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

Hey, rather neat! This operator--series correspondence is fun stuff! One last trick: using Pade approximants instead of series. series(Ddelta,h,6);

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

convert(%,ratpoly);

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small := expand( (80*(t->t)+9*h^2*(delta@@2))(D(f))(t) - 240/3*delta(f)(t) - 17/3*h^2*(delta@@3)(f)(t) );

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series( small, h, 8 ): simplify(%);

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

We have just seen our first "compact" finite difference formula, also known as a Pade method. Let's try another: series(Dln,h,6);

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

convert(%,ratpoly, 2,2 );

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Delta := f -> (t->(f(t+h)-f(t))/h);

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

small := expand((10*(t->t) + 12*h*Delta + 3*h^2*(Delta@@2))(D(f))(t) - (10*Delta + 7*h*(Delta@@2) + h^2*(Delta@@3)/3)(f)(t));

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series( small, h, 7 );

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Good, so we have a fifth-order accurate formula that uses four sample points...maybe we can do better?? And, wait a minute, that formula gives us a relation between the derivative of f, at three points, and the values of f, at four points. We'll have to do some work to sort that out!

Maxim #31, from http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/maxims.html Computational mathematics is mainly based on two ideas: Taylor series, and linear algebra. We don't have to be clever about this kind of thing, any more, though: we can simply use brute force. stencil := [[0,0],[h,0],[-h,0],[0,h],[0,-h]];

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The stencil is the description of where (which grid points) we know information at. N := nops( stencil );

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anszatz := add( b[i]*f(stencil[i][1],stencil[i][2]),i=1..N ) + b[N+1]*h^2*(D[1,1])(f)(-h,0) + b[N+2]*h^2*(D[1,1])(f)(h,0) ;

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The anszatz is a description of what we know at the stencil, and reflects the desired form of the finite difference approximation. In the way I have written it above, the anszatz has "zero weight". difexp := h*(D[1])(f)(0,0);

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The differential expression is what we wish to approximate by the anszatz. Now we take Taylor series of the difference between the anszatz and the differential expression, and set the coefficients to zero. zero := series( difexp - anszatz, h, N+3 );

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sol := solve( eqs, vars );

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formula := map( factor, subs( sol, anszatz ) );

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err := map(factor, series( difexp - formula, h, 7 ) );

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This worksheet procedure can be modified for whatever purpose you desire...in some sense this is all one needs. It's a bit clumsy, sort of like re-writing your Fortran every time you want to run the program again, and to make it work one has to ensure that all weights are correct, and that enough series terms are taken, but the final step (generation of the error coefficients) ensures that the results are correct.

The brute force approach discussed above has been turned into a reasonably easy to use program. read "C:/Data/Talks/CAIMS2005/findif10.mpl"; fd,err := FINDIF( f, [h], [[0],[h],[2*h]], [[[0]],[[0]],[[0]]], h*D[1](f)(h), 3, d );

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stencil := [[0,0,0], [Delta[x],0,0], [-Delta[x],0,0], [0,Delta[y],0], [0,-Delta[y],0], [0,0,Delta[z]], [0,0,-Delta[z]]];

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mask := [seq([[0,0,0],[1,0,0],[0,1,0],[0,0,1]],i=1..nops(stencil))];

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intrp := FINDIF( f, [Delta[x],Delta[y],Delta[z]], stencil, mask, Delta[x]*D[1](f)(a*Delta[x],b*Delta[y],c*Delta[z]), 3, d );

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Maxim #18: Symbolic computing is mainly useful when you want a symbolic answer. We have used this approach to generate a number of new formulae; new chiefly in the issue of closure at the boundary. I give a brief tutorial on the use of compact finite difference formulae here, and then discuss the refinements. First, let us look at a compact finite difference formula for the first derivative. restart; read "c:/data/talks/caims2005/jichao/first.mpl"; Digits := trunc(evalhf(Digits));

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

f := x -> sin(Pi*x)+cos(2*x);

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

df := (D)(f);

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

FDMP = First Derivative Matrix Pencil We use this program to have a look at a particular family of compact finite differences, generated using FINDIF in order for fast solution. Each row of the matrix pencil represents one invocation of a compact finite difference formula; the first row and the last row are special because they cannot involve points to the left (for the first row) or points to the right (for the last row), and special formulae were generated for those equations. 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M, C := FDMP( N, h, 0, 0, float[8] );

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Why such a peculiar M matrix? We have that M D(f) = C f, and so in order to find our derivatives we must solve the system. We see above that M is tridiagonal, which makes it easy...but in fact the first entry of M was chosen to make it even easier! 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 L,U :=LinearAlgebra:-LUDecomposition( M, output=['L','U'] );

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

2.0+sqrt(3.0); 1/(2.0+sqrt(3.0));

NiQtSSNtbkc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGKDY5US0zLjczMjA1MDgwNzZGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dGNS8lKnVuZGVybGluZUdGNS8lKnN1YnNjcmlwdEdGNS8lLHN1cGVyc2NyaXB0R0Y1LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYoLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GKC8lJ29wYXF1ZUdGNS8lK2V4ZWN1dGFibGVHRjUvJSlyZWFkb25seUdGNS8lKWNvbXBvc2VkR0Y1LyUqY29udmVydGVkR0Y1LyUraW1zZWxlY3RlZEdGNS8lLHBsYWNlaG9sZGVyR0Y1LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RigvJSptYXRoY29sb3JHRkAvJS9tYXRoYmFja2dyb3VuZEdGQy8lK2ZvbnRmYW1pbHlHRi8vJSxtYXRodmFyaWFudEdRJ25vcm1hbEYoLyUpbWF0aHNpemVHRjI3IyQiLypvdiEzMEtQISM4

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

The LU factoring of M can be done exactly