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check_ark.m
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check_ark.m
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function [qE,qI,q,qsE,qsI,qsA] = check_ark(cE,cI,AE,AI,bE,bI,tol,reportL,doPlot,box,mname,fname)
% Usage: [qE,qI,q,qsE,qsI,qsA] = check_ark(cE,cI,AE,AI,bE,bI,tol,reportL,doPlot,box,mname,fname)
%
% Checks the pair of Butcher tables given by
%
% cE | AE cI | AI
% -------- --------
% | bE | bI
%
% to determine the analytical order of accuracy for each method
% alone (qE and qI) as well as the combined ARK method order (q).
% This function uses the order condition equations from the article
% Sandu & Gunther, SINUM 53, 2015.
%
% We also compute the 'stage order' of the ERK, DIRK and ARK methods,
% respectively, through checking the Butcher simplifying assumption C(q)
% as prescribed in the article
% J.C. Butcher, "Implicit Runge-Kutta processes", Math Comp 18 (1964), pp 50-64.
%
% Inputs:
% cE, AE, bE -- ERK table
% cI, AI, bI -- DIRK table
% tol -- tolerance for checking order conditions (use negative
% or zero for default value)
% reportL -- integer flag denoting print level:
% >1 -> all results
% =1 -> final results
% else -> no results
% doPlot -- boolean flag denoting whether to create/save plot
% box -- [xl,xr,yl,yr] sub-region of the complex plane to
% use in plot (ignored if doPlot is false)
% mname -- string containing method name to insert into plot title
% fname -- string containing method name to insert into plot filenames
%
% Outputs:
% qE -- order of accuracy for ERK method
% qI -- order of accuracy for DIRK method
% q -- order of accuracy for combined ARK method
% qsE -- stage order of accuracy for ERK method
% qsI -- stage order of accuracy for DIRK method
% qsA -- stage order of accuracy for combined ARK method
%
%------------------------------------------------------------
% Programmer(s): Daniel R. Reynolds @ SMU
%------------------------------------------------------------
% Copyright (c) 2018, Southern Methodist University.
% All rights reserved.
% For details, see the LICENSE file.
%------------------------------------------------------------
% set tolerance on 'equality'
if (tol <= 0)
tol = 1e-8;
end
% check individual Butcher tables using single-table routine
BE = [cE, AE; 0, bE];
[qE,pE] = check_butcher(BE);
BI = [cI, AI; 0, bI];
[qI,pI,BsImp,AsImp,LsImp] = check_butcher(BI);
% check stage order of component and overall methods
[qsE, qsI, qsA] = check_ark_stage_order(cE,cI,AE,AI,tol);
% determine whether either method satisfies the 'stiffly accurate' property
expSA = true;
impSA = true;
for i=1:length(bE)
if (abs(bE(i) - AE(end,i)) > tol)
expSA = false;
break;
end
end
for i=1:length(bI)
if (abs(bI(i) - AI(end,i)) > tol)
impSA = false;
break;
end
end
% initialize failure flags, order of accuracy
failed = false;
failE = false;
failI = false;
failC = false;
q = -1;
% convert bE and bI to column vectors for these tests
bE = reshape(bE,length(bE),1);
bI = reshape(bI,length(bI),1);
% create cell arrays to streamline pairwise testing
bs = {bE,bI};
As = {AE,AI};
cs = {cE,cI};
% ensure row sum condition c(i) = sum(A(i,:)) is satisfied for each method
[s,tmp] = size(AE);
for i=1:s
tst = cE(i) - sum(AE(i,:));
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails row sum condition (i = %i, tst = %g)\n', i, tst);
end
end
tst = cI(i) - sum(AI(i,:));
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails row sum condition (i = %i, tst = %g)\n', i, tst);
end
end
end
if (~failed)
q = 0;
end
if (~failE)
qE = max(qE,0);
end
if (~failI)
qI = max(qI,0);
end
% check for first order
if (~failed || ~failE || ~failI)
tst = sum(bE)-1;
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 1st order condition (tst = %g)\n', tst);
end
end
tst = sum(bI)-1;
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 1st order condition (tst = %g)\n', tst);
end
end
if (reportL>1)
if (~failE), fprintf(' ERK method passes order 1 conditions\n'); end
if (~failI), fprintf(' DIRK method passes order 1 conditions\n'); end
end
if (~failed)
q = 1;
end
if (~failE)
qE = max(qE,1);
end
if (~failI)
qI = max(qI,1);
end
end
% check for second order
if (~failed || ~failE || ~failI)
tst = bE'*cE - 0.5;
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 2nd order condition (tst = %g)\n', tst);
end
end
tst = bI'*cI - 0.5;
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 2nd order condition (tst = %g)\n', tst);
end
end
tst = bE'*cI - 0.5;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 2nd order coupling condition A1 (tst = %g)\n', tst);
end
end
tst = bI'*cE - 0.5;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 2nd order coupling condition A2 (tst = %g)\n', tst);
end
end
if (reportL>1)
if (~failE), fprintf(' ERK method passes order 2 conditions\n'); end
if (~failI), fprintf(' DIRK method passes order 2 conditions\n'); end
if (~failC), fprintf(' ARK method passes order 2 coupling conditions\n'); end
end
if (~failed)
q = 2;
end
if (~failE)
qE = max(qE,2);
end
if (~failI)
qI = max(qI,2);
end
end
% check for third order
if (~failed || ~failE || ~failI)
tst = bE'*(cE.*cE) - (1/3);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 3nd order condition A (tst = %g)\n', tst);
end
end
tst = bI'*(cI.*cI) - (1/3);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 3rd order condition A (tst = %g)\n', tst);
end
end
tst = bE'*(AE*cE) - (1/6);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 3nd order condition B (tst = %g)\n', tst);
end
end
tst = bI'*(AI*cI) - (1/6);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 3nd order condition B (tst = %g)\n', tst);
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for ic1 = 1:2
c1 = cs{ic1};
for ic2 = 1:2
c2 = cs{ic2};
tst = b'*(c1.*c2) - (1/3);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 3rd order coupling condition A%i (tst = %g)\n', counter, tst);
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for iA = 1:2
A = As{iA};
for ic = 1:2
c = cs{ic};
tst = b'*(A*c) - (1/6);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 3rd order coupling condition B%i (tst = %g)\n', counter, tst);
end
end
end
end
end
if (reportL>1)
if (~failE), fprintf(' ERK method passes order 3 conditions\n'); end
if (~failI), fprintf(' DIRK method passes order 3 conditions\n'); end
if (~failC), fprintf(' ARK method passes order 3 coupling conditions\n'); end
end
if (~failed)
q = 3;
end
if (~failE)
qE = max(qE,3);
end
if (~failI)
qI = max(qI,3);
end
end
% check for fourth order
if (~failed || ~failE || ~failI)
tst = bE'*(cE.*cE.*cE) - (1/4);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 4th order condition A (tst = %g)\n', tst);
end
end
tst = bI'*(cI.*cI.*cI) - (1/4);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 4th order condition A (tst = %g)\n', tst);
end
end
tst = (bE.*cE)'*(AE*cE) - (1/8);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 4th order condition B (tst = %g)\n', tst);
end
end
tst = (bI.*cI)'*(AI*cI) - (1/8);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 4th order condition B (tst = %g)\n', tst);
end
end
tst = bE'*AE*(cE.*cE) - (1/12);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 4th order condition C (tst = %g)\n', tst);
end
end
tst = bI'*AI*(cI.*cI) - (1/12);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 4th order condition C (tst = %g)\n', tst);
end
end
tst = bE'*AE*AE*cE - (1/24);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 4th order condition D (tst = %g)\n', tst);
end
end
tst = bI'*AI*AI*cI - (1/24);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 4th order condition D (tst = %g)\n', tst);
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for ic1 = 1:2
c1 = cs{ic1};
for ic2 = 1:2
c2 = cs{ic2};
for ic3 = 1:2
c3 = cs{ic3};
tst = b'*(c1.*c2.*c3) - (1/4);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 4th order coupling condition A%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for ic1 = 1:2
c1 = cs{ic1};
for iA = 1:2
A = As{iA};
for ic2 = 1:2
c2 = cs{ic2};
tst = (b.*c1)'*(A*c2) - (1/8);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 4th order coupling condition B%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for iA = 1:2
A = As{iA};
for ic1 = 1:2
c1 = cs{ic1};
for ic2 = 1:2
c2 = cs{ic2};
tst = b'*A*(c1.*c2) - (1/12);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 4th order coupling condition C%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for iA1 = 1:2
A1 = As{iA1};
for iA2 = 1:2
A2 = As{iA2};
for ic = 1:2
c = cs{ic};
tst = b'*A1*A2*c - (1/24);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 4th order coupling condition D%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
if (reportL>1)
if (~failE), fprintf(' ERK method passes order 4 conditions\n'); end
if (~failI), fprintf(' DIRK method passes order 4 conditions\n'); end
if (~failC), fprintf(' ARK method passes order 4 coupling conditions\n'); end
end
if (~failed)
q = 4;
end
if (~failE)
qE = max(qE,4);
end
if (~failI)
qI = max(qI,4);
end
end
% check for fifth order
if (~failed || ~failE || ~failI)
tst = bE'*(cE.*cE.*cE.*cE) - (1/5);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 5th order condition A (tst = %g)\n', tst);
end
end
tst = bI'*(cI.*cI.*cI.*cI) - (1/5);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 5th order condition A (tst = %g)\n', tst);
end
end
tst = (bE.*cE.*cE)'*(AE*cE) - (1/10);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 5th order condition B (tst = %g)\n', tst);
end
end
tst = (bI.*cI.*cI)'*(AI*cI) - (1/10);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 5th order condition B (tst = %g)\n', tst);
end
end
tst = bE'*((AE*cE).*(AE*cE)) - (1/20);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 5th order condition C (tst = %g)\n', tst);
end
end
tst = bI'*((AI*cI).*(AI*cI)) - (1/20);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' DIRK method fails 5th order condition C (tst = %g)\n', tst);
end
end
tst = (bE.*cE)'*AE*(cE.*cE) - (1/15);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 5th order condition D (tst = %g)\n', tst);
end
end
tst = (bI.*cI)'*AI*(cI.*cI) - (1/15);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 5th order condition D (tst = %g)\n', tst);
end
end
tst = bE'*AE*(cE.*cE.*cE) - (1/20);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 5th order condition E (tst = %g)\n', tst);
end
end
tst = bI'*AI*(cI.*cI.*cI) - (1/20);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' DIRK method fails 5th order condition E (tst = %g)\n', tst);
end
end
tst = (bE.*cE)'*AE*AE*cE - (1/30);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 5th order condition F (tst = %g)\n', tst);
end
end
tst = (bI.*cI)'*AI*AI*cI - (1/30);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' DIRK method fails 5th order condition F (tst = %g)\n', tst);
end
end
tst = -(1/40);
for i=1:s
for j=1:s
for k=1:s
tst = tst + bE(i)*AE(i,j)*cE(j)*AE(j,k)*cE(k);
end
end
end
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 5th order condition G (tst = %g)\n', tst);
end
end
tst = -(1/40);
for i=1:s
for j=1:s
for k=1:s
tst = tst + bI(i)*AI(i,j)*cI(j)*AI(j,k)*cI(k);
end
end
end
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' DIRK method fails 5th order condition G (tst = %g)\n', tst);
end
end
tst = bE'*AE*AE*(cE.*cE) - (1/60);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 5th order condition H (tst = %g)\n', tst);
end
end
tst = bI'*AI*AI*(cI.*cI) - (1/60);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' DIRK method fails 5th order condition H (tst = %g)\n', tst);
end
end
tst = bE'*AE*AE*AE*cE - (1/120);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 5th order condition I (tst = %g)\n', tst);
end
end
tst = bI'*AI*AI*AI*cI - (1/120);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' DIRK method fails 5th order condition I (tst = %g)\n', tst);
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for ic1 = 1:2
c1 = cs{ic1};
for ic2 = 1:2
c2 = cs{ic2};
for ic3 = 1:2
c3 = cs{ic3};
for ic4 = 1:2
c4 = cs{ic4};
tst = b'*(c1.*c2.*c3.*c4) - (1/5);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 5th order coupling condition A%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for ic1 = 1:2
c1 = cs{ic1};
for ic2 = 1:2
c2 = cs{ic2};
for iA = 1:2
A = As{iA};
for ic3 = 1:2
c3 = cs{ic3};
tst = (b.*c1.*c2)'*(A*c3) - 1/10;
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 5th order coupling condition B%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for iA1 = 1:2
A1 = As{iA1};
for ic1 = 1:2
c1 = cs{ic1};
for iA2 = 1:2
A2 = As{iA2};
for ic2 = 1:2
c2 = cs{ic2};
tst = b'*((A1*c1).*(A2*c2)) - (1/20);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 5th order coupling condition C%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for ic1 = 1:2
c1 = cs{ic1};
for iA = 1:2
A = As{iA};
for ic2 = 1:2
c2 = cs{ic2};
for ic3 = 1:2
c3 = cs{ic3};
tst = (b.*c1)'*A*(c2.*c3) - (1/15);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 5th order coupling condition D%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for iA = 1:2
A = As{iA};
for ic1 = 1:2
c1 = cs{ic1};
for ic2 = 1:2
c2 = cs{ic2};
for ic3 = 1:2
c3 = cs{ic3};
tst = b'*A*(c1.*c2.*c3) - (1/20);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 5th order coupling condition E%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for ic1 = 1:2
c1 = cs{ic1};
for iA1 = 1:2
A1 = As{iA1};
for iA2 = 1:2
A2 = As{iA2};
for ic2 = 1:2
c2 = cs{ic2};
tst = (b.*c1)'*A1*A2*c2 - (1/30);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 5th order coupling condition F%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for iA1 = 1:2
A1 = As{iA1};
for ic1 = 1:2
c1 = cs{ic1};
for iA2 = 1:2
A2 = As{iA2};
for ic2 = 1:2
c2 = cs{ic2};
tst = -(1/40);
for i=1:s
for j=1:s
for k=1:s
tst = tst + b(i)*A1(i,j)*c1(j)*A2(j,k)*c2(k);
end
end
end
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 5th order coupling condition G%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for iA1 = 1:2
A1 = As{iA1};
for iA2 = 1:2
A2 = As{iA2};
for ic1 = 1:2
c1 = cs{ic1};
for ic2 = 1:2
c2 = cs{ic2};
tst = b'*A1*A2*(c1.*c2) - (1/60);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 5th order coupling condition H%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
end
counter=0;
for ib = 1:2
b = bs{ib};
for iA1 = 1:2
A1 = As{iA1};
for iA2 = 1:2
A2 = As{iA2};
for iA3 = 1:2
A3 = As{iA3};
for ic = 1:2
c = cs{ic};
tst = b'*A1*A2*A3*c - (1/120);
counter = counter+1;
if (abs(tst) > tol)
failed = true;
failC = true;
if (reportL>1)
fprintf(' ARK method fails 5th order coupling condition I%i (tst = %g)\n', counter, tst);
end
end
end
end
end
end
end
if (reportL>1)
if (~failE), fprintf(' ERK method passes order 5 conditions\n'); end
if (~failI), fprintf(' DIRK method passes order 5 conditions\n'); end
if (~failC), fprintf(' ARK method passes order 5 coupling conditions\n'); end
end
if (~failed)
q = 5;
end
if (~failE)
qE = max(qE,5);
end
if (~failI)
qI = max(qI,5);
end
end
% check for sixth order
if (~failed || ~failE || ~failI)
tst = bE'*(cE.*cE.*cE.*cE.*cE) - (1/6);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 6th order condition A (tst = %g)\n', tst);
end
end
tst = bI'*(cI.*cI.*cI.*cI.*cI) - (1/6);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 6th order condition A (tst = %g)\n', tst);
end
end
tst = (bE.*cE.*cE.*cE)'*AE*cE - (1/12);
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 6th order condition B (tst = %g)\n', tst);
end
end
tst = (bI.*cI.*cI.*cI)'*AI*cI - (1/12);
if (abs(tst) > tol)
failed = true;
failI = true;
if (reportL>1)
fprintf(' DIRK method fails 6th order condition B (tst = %g)\n', tst);
end
end
tst = -(1/24);
for i=1:s
for j=1:s
for k=1:s
tst = tst + bE(i)*cE(i)*AE(i,j)*cE(j)*AE(j,k)*cE(k);
end
end
end
if (abs(tst) > tol)
failed = true;
failE = true;
if (reportL>1)
fprintf(' ERK method fails 6th order condition C (tst = %g)\n', tst);
end
end
tst = -(1/24);
for i=1:s
for j=1:s