A Comparision of Time Marching Methods Coupled with a Higher-Order TVD Scheme for Time-Dependent Advection Equation.

Abstract

Time marching methods coupled with a third order upwind TVD scheme are compared from the viewpoints of accuracy, memory requirement and computational cost. The methods examined are the second and the third order Runge-Kutta presented by Shu and Osher, the explicit Euler, the second order Adams-Bashforth, the implicit Euler and the Crank-Nicolson methods. Benchmark problems on time-dependent linear advective transport are solved using these methods. As a result, the Shu and Osher's Runge-Kutta methods of the second and the third order are recommended. The explicit Euler method suffers from oversteepening of the gradient due to numerical diffusion with a negative coefficient. The Adams-Bashforth method is inferior to the Runge-Kutta method in accuracy and memory requirement. The Crank-Nicolson method produces solutions as accurate as the Runge-Kutta method, but requires much CPU time due to the nonlinear feature of the TVD flux limiter.