3
$\begingroup$

I want to solve a system of Pde (2D) reaction diffusion type using NDSolve

enter image description here

whose boundary conditions are

enter image description here

and the initial conditions are

enter image description here

or

enter image description here

I thought of the following code

(*parameters*) L = 5; T = 10; (*system of nonlinear PDE*) pde = {D[N1[t, x, y], t] == D[N1[t, x, y], x, x] + D[N1[t, x, y], y, y] + (1 - N1[t, x, y] - 0.5 N2[t, x, y]) N1[t, x, y], D[N2[t, x, y], t] == D[N2[t, x, y], x, x] + D[N2[t, x, y], y, y] + (1 - N2[t, x, y] - 0.5 N1[t, x, y]) N2[t, x, y]}; (*periodic boundary condition*) bc = {N1[t, -L, y] == N1[t, L, y], N1[t, x, -L] == N1[t, x, L], N2[t, -L, y] == N2[t, L, y], N2[t, x, -L] == N2[t, x, L]}; (*initial condition*) ic = {N1[0, x, y] == If[-0.5 <= x <= 1 && -0.5 <= y <= 1, 1, 0], N2[0, x, y] == If[-0.5 <= x <= 1 && -0.5 <= y <= 1, 0, 1]}; eqns = Flatten@{pde, bc, ic}; {N1, N2} = NDSolve[eqns, {N1, N2}, {t, 0, T}, {x, -L, L}, {y, -L, L}, Method -> {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid"}}]

However, the following errors appear.

enter image description here

Also, when constructing the plot (by DensityPlot), I verify that there is a failure in the initial condition, as shows

enter image description here

Can someone help me?

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ 1. You need e.g. {solN1, solN2} = NDSolveValue[…… instead of {N1, N2} = NDSolve[…. 2. Try Plot3D instead of DensityPlot, or setting a larger PlotPoints for DensityPlot. $\endgroup$ Commented Nov 30, 2017 at 2:40

1 Answer 1

Reset to default
7
$\begingroup$

As @xzczd alluded to, your code is almost correct. The message NDSolve::mxsst is more of a note than an error or even a warning.

Here's a working version because I love reaction-diffusion equations (this appears to be a diffusive Lotka-Volterra competition model, parameterized so that the species can stably coexist). I used a diffusion coefficient d=0.002, to see the traveling wave better, increased the number of grid points, and fixed your initial conditions to match the description.

(*parameters*) L = 5; T = 100; d = 0.002; (*system of nonlinear PDE*) pde = { D[N1[t, x, y], t] == d*(D[N1[t, x, y], x, x] + D[N1[t, x, y], y, y]) + (1 - N1[t, x, y] - 0.5 N2[t, x, y]) N1[t, x, y], D[N2[t, x, y], t] == d*(D[N2[t, x, y], x, x] + D[N2[t, x, y], y, y]) + (1 - N2[t, x, y] - 0.5 N1[t, x, y]) N2[t, x, y] }; (*periodic boundary condition*) bc = {N1[t, -L, y] == N1[t, L, y], N1[t, x, -L] == N1[t, x, L], N2[t, -L, y] == N2[t, L, y], N2[t, x, -L] == N2[t, x, L]}; (*initial condition*) ic = {N1[0, x, y] == If[-0.5 <= x <= 0.5 && -0.5 <= y <= 0.5, 1, 0], N2[0, x, y] == If[-0.5 <= x <= 0.5 && -0.5 <= y <= 0.5, 0, 1]}; eqns = Flatten@{pde, bc, ic}; sol = NDSolve[eqns, {N1, N2}, {t, 0, T}, {x, -L, L}, {y, -L, L}, Method -> {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid", "MaxPoints" -> 200}}][[1]];

Here's a movie:

img

Improve this answer
$\endgroup$

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.