HeatEquation.md

Solving the Heat Equation using DiffEqOperators

In this tutorial we will solve the famous heat equation using the explicit discretization on a 2D space x time grid. The heat equation is:

$$\frac{\partial u}{\partial t} - \frac{{\partial}^2 u}{\partial x^2} = 0$$

For this example we consider a Dirichlet boundary condition with the initial distribution being parabolic. Since we have fixed the value at boundaries (in this case equal), after a long time we expect the 1D rod to be heated in a linear manner.

using DiffEqOperators, DifferentialEquations, Plots
x = collect(-pi : 2pi/511 : pi);
u0 = -(x .- 0.5).^2 .+ 1/12;
A = DerivativeOperator{Float64}(2,2,2pi/511,512,:Dirichlet,:Dirichlet;BC=(u0[1],u0[end]));

Now solving equation as an ODE we have:

prob1 = ODEProblem(A, u0, (0.,10.));
sol1 = solve(prob1, dense=false, tstops=0:0.01:10);
# try to plot the solution at different time points using
plot(x, [sol1(i) for i in 0:1:10])

Note: Many a times the solver may inform you that the solution is unstable. This problem is usually handled by changing the solver algorithm. Your best bet might be the CVODE_BDE() algorithm from the OrdinaryDiffEq.jl suite.