MATLAB: Spatial discretization has failed. Discretization supports only parabolic and elliptic equations, with flux term involving spatial derivative.

pdepespatial discretion has failed

I get this error in writing the code…. The code I wrote is as follows
function pdeop global A void G L Usg Usl Rhog Rhol Kos Kot aeff Ugeff Uleff hl a; m = 0; x = linspace(0,1.5,31); t = linspace(0,3600,3601); sol = pdepe(m,@pdeoppde,@pdeopic,@pdeopbc,x,t); u1 = sol(:,:,1); u2 = sol(:,:,2); u3=sol(:,:,3); u4=sol(:,:,4);
% ————————————————————– function [c,f,s] = pdeoppde(x,t,u,DuDx) c = [A*void*(1-hl)*Rhog;A*void*hl*Rhol;A*void*(1-hl)*Rhog;A*void*hl*Rhol]; f = [-G;L;-G;L] .*u; Cs=Rhog*Kos*aeff*A*sqrt(Rhog/(Rhol-Rhog))*(Ugeff/Uleff); Ct=Rhog*Kot*aeff*A*sqrt(Rhog/(Rhol-Rhog))*(Ugeff/Uleff); ysstar=-4.2913*u(2)^4+11.791*u(2)^3-12.01*u(2)^2+5.508*u(2); ytstar=0.0023*exp(6.0058*u(4)); M=Cs*(ysstar-u(1)); N=Ct*(ytstar-u(3)); s = [M;-M;N;-N]; end % ————————————————————– function u0 = pdeopic(x) u0 = [0;0.5;0;0.5]; end % ————————————————————– function[pl,ql,pr,qr] = pdeopbc(xl,ul,xr,ur,t) pl = [ul(1);0;ul(3);0]; ql = [0;0;0;0]; pr = [ur(1)-0.5;0;ur(3)-0.5;0]; qr = [0;0;0;0]; end end
Can anyone please suggest me a solution?

Best Answer

I see two problems.
First, your boundary conditions are incorrectly defined. They should be:
pl = [ul(1);ul(2);ul(3);ul(4)]; 
pr = [ur(1);ur(2)-0.5;ur(3);ur(4)-0.5];
(ql and qr are correct). All four of your equations are hyperbolic which pdepe is not designed to handle. But, you have added some artificial diffusion (the small second derivative terms). That is a good approach but because of the negative signs on some of the coefficients you have de-stabilized the system instead stabilizing it. Try this for the vector of diffusion coefficients, instead:
eps=.001;
[eps eps eps eps]'
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