Truncation and accumulated errors in wave propagation

Abstract

The approximation of the truncation and accumulated errors in the numerical solution of a linear initial-valued partial differential equation problem can be established by using a semidiscretized scheme. This error approximation is observed as a lower bound to the errors of a finite difference scheme. By introducing a modified von Neumann solution, this error approximation is applicable to problems with variable coefficients. To seek an in-depth understanding of this newly established error approximation, numerical experiments were performed to solve the hyperbolic equation ∂U ∂t = −C 1(x)C 2(t) ∂U ∂x , with both continuous and discontinuous initial conditions. We studied three cases: (1) C 1( x)= C 0 and C 2( t)=1; (2) C 1( x)= C 0 and C 2( t= t; and (3) C 1(x)=1+( solx a ) 2 and C 2( t)= C 0. Our results show that the errors are problem dependent and are functions of the propagating wave speed. This suggests a need to derive problem-oriented schemes rather than the equation-oriented schemes as is commonly done. Furthermore, in a wave-propagation problem, measurement of the error by the maximum norm is not particularly informative when the wave speed is incorrect.