Support Operators Method for the Diffusion Equation in Multiple Materials

A second-order finite difference scheme for the solution of the diffusion equation on non-uniform meshes is implemented. The method allows the heat conductivity to be discontinuous. The algorithm is formulated on a one dimensional mesh and is derived using the support operators method. A key component of the derivation is that the discrete analog of the flux operator is constructed to be the negative adjoint of the discrete divergence, in an inner product that is a discrete analog of the continuum inner product. The resultant discrete operators in the fully discretized diffusion equation are symmetric and positive definite. The algorithm is generalized to operate on meshes with cells which have mixed material properties. A mechanism to recover intermediate temperature values in mixed cells using a limited linear reconstruction is introduced. The implementation of the algorithm is verified and the linear reconstruction mechanism is compared to previous results for obtaining new material temperatures.