pde1d

1D Partial Differential Equation Solver for MATLAB and Octave

pde1d solves systems of partial differential equations (PDE) in a single spatial variable and time. The input is mostly compatible with the MATLAB function pdepe. Many pdepe examples will work with pde1d with only small changes.

However, pde1d contains several enhancements which make it substantially more powerful than pdepe. Specifically, pde1d allows any number of ordinary differential equations (ODE) to be coupled to the system of PDE. One use of these ODE, for example, is to allow more complex boundary conditions at the two ends of the PDE domain. Another benefit of pde1d relative to pdepe is improved performance, particularly when many mesh points are required for a converged solution. Third, pde1d allows advanced users to specify the order of the approximation functions in the spatial domain.

Two capabilities of pdepe are not currently supported by pde1d. pde1d does not allow complex coefficients. Also, when the PDE is defined in a cylindrical or spherical coordinate system and the left end of the domain starts at zero, pdepe uses special approximation functions to account for the singularity at this point; pde1d does not.

Several examples and basic documentation are included in the user guide. An excellent introduction to solving PDE with the pdepe function is Professor Howard's note, Partial Differential Equations in MATLAB 7.0. His examples, modified for pde1d, can be found in the examples directory.

Build Instructions

pde1d is written in C++, is built using the CMake build system, and relies on the following third-party libraries. The IDA library from SUNDIALS is used for solution of the differential-algebraic equations. At least version 3 of Sundials is required. The Eigen C++ matrix class library, is used throughout the code. By default, pde1d uses the Eigen sparse LU solver to solve the linear algebraic system in SUNDIALS. However, if SUNDIALS has been built with the optional KLU sparse solver from SuiteSparse, pde1d can also be built to support this option. In this case the SuiteSparse libraries are required. There is also a small dependency on the Boost C++ string package. At least version 3.0 of CMake is required.

The first step in building is to download and unpack the pde1d source into a local directory. That local directory will be referred to as <path_to_pde1d_source> below. A second directory, referred to as the build directory, must then be created.

Linux

When all of the required libraries are installed in "standard" locations, pde1d can be built from the command line by invoking the following commands from the build directory:

cmake -DCMAKE_INSTALL_PREFIX=<somewhere_on_matlab_or_octave_path> \
  <path_to_pde1d_source>
make

where <somewhere_on_matlab_or_octave_path> is a directory where Matlab or Octave can find it.

By default, the CMakeLists.txt file assumes that pde1d is being built for Octave. To build for Matlab, the PSE CMake variable must be set as shown below:

cmake  -DPSE=Matlab \
 -DCMAKE_INSTALL_PREFIX=<somewhere_on_matlab_or_octave_path> \
   <path_to_pde1d_source>
make

Only the latest Linux distributions have installable packages for the required Sundials libraries. So it is often necessary to download the source code from the Sundials site and follow their installation procedure (also based on CMake) before proceeding with the pde1d build. If Sundials is installed in a non-standard location, it can be selected as follows in the pde1d build:

cmake  -DPSE=Matlab \
 -DCMAKE_INSTALL_PREFIX=<somewhere_on_matlab_or_octave_path>  \
  -DSUNDIALS_ROOT=<path_to_sundials> <path_to_pde1d_source>
make

Windows

The easiest way to run CMake on Windows platforms is using the graphical interface cmake-gui that is part of the Windows CMake distribution. The source and build directories along with the necessary CMake configuration variables can be set in the GUI. Because Windows doesn't have standard locations for the installation of third-party development libraries, the following CMake variables typically must be set: EIGEN_ROOT, SUNDIALS_ROOT, CMAKE_INSTALL_PREFIX. CMake supports several native build systems on Windows including MS Visual Studio. After CMake generates the Visual Studio solution, it can be opened and built from Visual Studio using the "Build Solution" menu selection.