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driver_p2.m
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% driver for stiff brusselator test problem:
% u' = a - (w+1)*u + u^2*v,
% v' = w*u - u^2*v,
% w' = (b-w)/ep - u*w,
% where u(0) = 1.2, v(0) = 3.1 and w(0) = 3, with prameters a=1,
% b=3.5 and ep=5e-6. We evaluate over the time interval [0,10].
%
% Daniel R. Reynolds
% Department of Mathematics
% Southern Methodist University
% March 2017
% All Rights Reserved
clear
% set problem parameters
a = 1;
b = 3.5;
global ep;
ep = 1e-3;
fn = @(t,y) [a - (y(3)+1)*y(1) + y(1)*y(1)*y(2);
y(3)*y(1) - y(1)*y(1)*y(2);
(b-y(3))/ep - y(3)*y(1)];
fe = @(t,y) [a - (y(3)+1)*y(1) + y(1)*y(1)*y(2);
y(3)*y(1) - y(1)*y(1)*y(2);
-y(3)*y(1)];
fi = @(t,y) [0; 0; (b-y(3))/ep];
Jn = @(t,y) [-(y(3)+1) + 2*y(1)*y(2), y(1)*y(1), -y(1);
y(3) - 2*y(1)*y(2), -y(1)*y(1), y(1);
-y(3), 0, -1/ep - y(1)];
Ji = @(t,y) [0, 0, 0; 0, 0, 0; 0, 0, -1/ep];
Es = @EStab_p2;
Tf = 10;
tout = linspace(0,Tf,100);
hmin = 1e-7;
hmax = 1.0;
rtol = 1e-4;
atol = 1e-14*ones(3,1);
u0 = 1.2;
v0 = 3.1;
w0 = 3;
Y0 = [u0; v0; w0];
% plot "true" solution
opts = odeset('RelTol',1e-12, 'AbsTol',atol,'InitialStep',hmin/10, 'MaxStep',hmax);
[t,Ytrue] = ode15s(fn, tout, Y0, opts);
figure()
plot(tout,Ytrue)
xlabel('t','FontSize',12), ylabel('y','FontSize',12)
title('Brusselator ODE test','FontSize',14)
set(gca,'FontSize',12)
print('-dpng','brusselator')
% run with an embedded diagonally-implicit RK method
mname = 'Cash(5,3,4)-SDIRK';
B = butcher(mname); s = numel(B(1,:))-1;
fprintf('\nRunning with DIRK integrator: %s (order = %i)\n',mname,B(s+1,1))
[t,Y,ns,nl,~] = solve_DIRK(fn, Jn, tout, Y0, B, rtol, atol, hmin, hmax, hmin);
err_max = max(max(abs(Y'-Ytrue)));
err_rms = sqrt(sum(sum((Y'-Ytrue).^2))/numel(Y));
fprintf('Accuracy/Work Results:\n')
fprintf(' maxerr = %.5e, rmserr = %.5e\n',err_max, err_rms);
fprintf(' steps = %i (stages = %i), linear solves = %i\n',ns,ns*s,nl);
% run with a non-embedded diagonally-implicit RK method
mname = 'Cooper4-ESDIRK';
B = butcher(mname); s = numel(B(1,:))-1;
fprintf('\nRunning with DIRK integrator: %s (order = %i)\n',mname,B(s+1,1))
[t,Y,ns,nl,~] = solve_DIRK(fn, Jn, tout, Y0, B, rtol, atol, hmin, hmax, hmin);
err_max = max(max(abs(Y'-Ytrue)));
err_rms = sqrt(sum(sum((Y'-Ytrue).^2))/numel(Y));
fprintf('Accuracy/Work Results:\n')
fprintf(' maxerr = %.5e, rmserr = %.5e\n',err_max, err_rms);
fprintf(' steps = %i (stages = %i), linear solves = %i\n',ns,ns*s,nl);
% run with a fully-implicit RK method
mname = 'LobattoIIIC-3-4-IRK';
B = butcher(mname); s = numel(B(1,:))-1;
fprintf('\nRunning with IRK integrator: %s (order = %i)\n',mname,B(s+1,1))
[t,Y,ns,nl,~] = solve_IRK(fn, Jn, tout, Y0, B, rtol, atol, hmin, hmax, hmin);
err_max = max(max(abs(Y'-Ytrue)));
err_rms = sqrt(sum(sum((Y'-Ytrue).^2))/numel(Y));
fprintf('Accuracy/Work Results:\n')
fprintf(' maxerr = %.5e, rmserr = %.5e\n',err_max, err_rms);
fprintf(' steps = %i (stages = %i), linear solves = %i\n',ns,ns*s,nl);
% run with an embedded ARK method
mname1 = 'ARK4(3)6L[2]SA-ERK';
Be = butcher(mname1); s = numel(Be(1,:))-1;
mname2 = 'ARK4(3)6L[2]SA-ESDIRK';
Bi = butcher(mname2);
fprintf('\nRunning with ARK integrator: %s/%s (order = %i)\n',...
mname1,mname2,Be(s+1,1))
[t,Y,ns,nl] = solve_ARK(fe, fi, Ji, tout, Y0, Be, Bi, rtol, atol, hmin, hmax, hmin);
err_max = max(max(abs(Y'-Ytrue)));
err_rms = sqrt(sum(sum((Y'-Ytrue).^2))/numel(Y));
fprintf('Accuracy/Work Results:\n')
fprintf(' maxerr = %.5e, rmserr = %.5e\n',err_max, err_rms);
fprintf(' steps = %i (stages = %i), linear solves = %i\n',ns,ns*s,nl);
% run with an embedded explicit RK method
mname = 'Fehlberg-ERK';
B = butcher(mname); s = numel(B(1,:))-1;
fprintf('\nRunning with ERK integrator: %s (order = %i)\n',mname,B(s+1,1))
[t,Y,ns,~] = solve_ERK(fn, Es, tout, Y0, B, rtol, atol, hmin, hmax, hmin);
err_max = max(max(abs(Y'-Ytrue)));
err_rms = sqrt(sum(sum((Y'-Ytrue).^2))/numel(Y));
fprintf('Accuracy/Work Results:\n')
fprintf(' maxerr = %.5e, rmserr = %.5e\n',err_max, err_rms);
fprintf(' steps = %i (stages = %i)\n',ns,ns*s);
% run with a non-embedded explicit RK method
mname = 'ERK-4-4';
B = butcher(mname); s = numel(B(1,:))-1;
fprintf('\nRunning with ERK integrator: %s (order = %i)\n',mname,B(s+1,1))
[t,Y,ns,~] = solve_ERK(fn, Es, tout, Y0, B, rtol, atol, hmin, hmax, hmin);
err_max = max(max(abs(Y'-Ytrue)));
err_rms = sqrt(sum(sum((Y'-Ytrue).^2))/numel(Y));
fprintf('Accuracy/Work Results:\n')
fprintf(' maxerr = %.5e, rmserr = %.5e\n',err_max, err_rms);
fprintf(' steps = %i (stages = %i)\n',ns,ns*s);
% end of script
%------------------------- Utility routines -------------------------%
function dt = EStab_p2(t, y)
% compute the Jacobian ODE RHS
global ep;
J = [-(y(3)+1) + 2*y(1)*y(2), y(1)*y(1), -y(1);
y(3) - 2*y(1)*y(2), -y(1)*y(1), y(1);
-y(3), 0, -1/ep - y(1)];
% determine the largest eigenvalue magnitude
lam = max(abs(eig(J)));
% assume explicit stability region includes Euler stability region
% (this assumes that the eigenvalue is in fact negative).
dt = 1/lam;
% end function
end