# Solving a system of nonlinear PDEs

I'm trying to solve a system of coupled PDEs as described below. The written code works when executed in Mathematica 10.4.

However, as I change the exponent of the dependent variable u[t,x] in the 2nd pde (the aim is nexp=0.6), no solution is reached. Do you have an idea as to how I have to change the code?

horst

nexp = 1;(*[-] exponent for u[t,x] in 2nd pde*)

(* Dimensionless Parameters*)
Pem = 1000;
beta1 = 160000;
beta2 = 0.005;
beta3 = 0.7;

(*Integration Parameters*)
tMax = 300000; (*[-] Int. time.*)

(*Solve PDE*)
pde = {D[u[t, x], t] + D[u[t, x], x] - (1/Pem)*D[u[t, x], x, x] + beta1*D[v[t, x], t] == 0,
D[v[t, x], t] == beta2*(beta3*u[t, x]^nexp - v[t, x])};

bc = {u[0, x] == Exp[-10^3*x],
u[t, 0] == 1, (*Derivative[0,1][u][t,0]==Pem*(u[t,0]-1),*)
Derivative[0, 1][u][t, 1] == 0,
v[0, x] == 0};

sol = NDSolve[{pde, bc}, {u, v}, {t, 0, tMax}, {x, 0, 1},
Method -> {"PDEDiscretization" -> {"MethodOfLines",     "SpatialDiscretization" -> {"TensorProductGrid", "MinPoints" -> 10}}}];

(*Plotting/Exporting*)
Plot3D[Evaluate[{u[t, x]} /. sol], {x, 0, 1}, {t, 0, tMax}]
Plot[Evaluate[u[t, x] /. sol /. x -> 1], {t, 0, tMax}, PlotRange -> All]


Seems to be a stiff non-linear problem. NDSolve is actually time integration this though very slowly. Setting the time integration to use "StiffnessSwitching" helps a bit:

 Monitor[
sol = NDSolve[{pde, bc}, {u, v}, {t, 0, tMax}, {x, 0, 1},
Method -> {"PDEDiscretization" -> {"MethodOfLines",
"SpatialDiscretization" -> {"TensorProductGrid",
"MinPoints" -> 10}},
"TimeIntegration" -> "StiffnessSwitching"
}, EvaluationMonitor :> (monitor =
Row[{"t = ", CForm[t]}])], monitor]

• Thank you for the hint. Is there an other way for speed-up (e.g. using an other method rather than MOL)? – Horst Sep 7 '16 at 14:58
• I do not think you will get around the MOL, try playing with the time integration and see if you can find a better suited one., Try also to set the "MaxPoints" to 10 and see if it solves it without complaints. – user21 Sep 7 '16 at 15:14