Modified mass matrices and positivity preservation for hyperbolic and parabolic PDEs

SUMMARY Modifications to the standard finite element mass matrix are considered with the aim of preserving the positivity of the discrete solution. The approach is used in connection with calculating the initial time derivative values for parabolic equations and in connection with nonlinear Petrov-Galerkin schemes for hyperbolic equations in one space dimension. The extension of the ideas to unstructured meshes in two and three space dimensions is indicated. Copyright c 2000 John Wiley & Sons, Ltd. There are many situations in the numerical solution of partial differential equations in which the computed solution values should, on physical grounds, remain non-negative. One the simplest examples is that of the simple advection equation with non-negative initial data while other cases are those of concentrations of chemical compounds in reacting flow calculations. In the latter case preserving positivity is essential to avoid the numerical calculation becoming meaningless. Consider the solution of the advection equation with appropriate initial and boundary condition by using the standard Galerkin method with linear basis (hat) functions φ i x on a uniformly spaced mesh x i i 1 N to get