ON THE DYNAMICS OF SOME DISCRETISATIONS OF CONVECTION-DIFFUSION EQUATIONS

Numerical discretisations of differential equations which model physic al processes can possess dynamics guile different from that of dhe equ ations themselves (see [3] and references therein). Recently the empha sis of study has been on the the dynamics of numerical discretisations for Ordinary Differential Equations (ODEs). For Partial Differential Equations (PDEs) using a method of lines approach the situation is mor e complex. First, dhe spatial discretisation may introduce dynamics no t present in the original equations; second, the solution of the resul ting system of ODEs is open to the modified dynamics of the ODE solver used. These two effects may interact in a complex manner. In this sho rt paper we summarise some results of our recent work on the dynamics of discretisations of convection-diffusion equations using Total Varia tion Diminishing (TVD) schemes and adaptive grid techniques. Results u sing classical spatial differencing (non-TVD) are presented in [1].