Finite difference method

Abstract: In this paper we show that the total variation diminishing (TVD) finite difference scheme which was analysed by Sweby [8] can be interpreted as a Lax—Wendroff scheme plus an upwind weighted artificial dissipation term. We then show that if we choose a particular flux limiter and remove the requirement for upwind weighting, we obtain an artificial dissipation term which is based on the theory of TVD schemes, which does not contain any problem dependent parameters and which can be added to existing MacCormack method codes. Finally, we conduct numerical experiments to examine the performance of this new method.

Abstract: The technique of Richardson extrapolation, which has previously been used on time-independent problems, is extended so that it can also be used on time-dependent problems. The technique presented is completed in the sense that the extrapolated solution is calculated at all spatial grid nodes which coincide with nodes of the finest grid considered. Numerical examples are presented when the technique is applied to the Lax–Wendroff and Crank–Nicholson finite difference schemes which are used to approximate solutions to the convection–diffusion equation. The examples show that extrapolation can be an easy and efficient way in which to produce accurate numerical solutions to time-dependent problems. © 1997 John Wiley & Sons, Ltd.

Abstract: The finite element method, using smooth splines as basis functions, applied to the model problem $u_t = cu_x $ with periodic data generates a differential-difference equation whose phase error is closely estimated and compared with the phase error of both explicit and high order implicit centered differencing. We also compute and compare the minimum work required to obtain a fixed error for several fully discrete schemes.