1dpme

One dimensional Porous Medium Equation (PME) solved with a moving mesh approach described in the PhD thesis

A moving mesh finite element method for the numerical solution of partial differential equations and systems.

which can be found here.

The Porous Medium Equation is described by the non-linear partial differential equation

u_t = ∇.(u^m ∇u)

and admits radial self-similar solutions of the form

u(r,t) = (1/lambda^d)*(1-(r/(r0*lambda))^2)^(1/m)

where

r0^d = Q*gamma(1/m + 1/d + 1)/(gamma(d/2)*gamma(1/m + 1))
t0 = ((r0^2)*m)/(2*(d*m+2))
lambda = (t/t0)^(1/(d*m+2))

Here Q is the total mass of the solution gamma is the gamma function and d is the dimension of the problem.

Numerical Solution

The one-dimensional Porous Medium Equation is solved using a moving mesh method which uses the monitor function M=u(x,t) in the moving mesh equations for the mesh velocity. The mesh is advanced forwards in time using a forward Euler time-stepping scheme. The solution to the PME is obtained as a reconstruction step by considering conservation of the monitor function. All the moving mesh equations are solved using linear finite elements.

Building and Developing

Developing locally requires Docker for Windows. Run the command

docker build -t 1dpme .

to build the docker image which pulls the gcc base image containing gfortran and maps the source code into the container.

Then run image with the command:

docker run -i -t -v /f/git/src/github.com/bvwells/1dpme:/app 1dpme

This command maps the local workspace into the running image so any changes made in the running image will be reflected on the local workspace.

Within the running image generate the make files for the release version by running the command:

cmake .

To build the debug version of the code run the command:

cmake -DCMAKE_BUILD_TYPE=Debug

Build the executable by running the command:

make

Running

The program takes the file variables.data as input to the simulation. The program can be run from the base of the repo with the command:

./bin/1dpme.exe

The program outputs the mesh and solution over time into the files solutionXXX.m. The variables for the solution are written to the file variables.m.

Plotting Solution

The output from the simulation can be plotted in Octave by running the plotting file plot_solution.m in the root of the repo.