Closed Form Solution and Equivalent Equation Approximation of Linear Advection by a Non Dissipative Second Order Scheme for Step Initial Conditions

Abstract

We analyse the solution of the linear advection equation on a uniform mesh by a non dissipative second order scheme for discontinuous initial condition. These schemes are known to generate parasitic oscillations in the vicinity of the discontinuity. An approximate way to predict these oscillations is provided by the equivalent equation method. More specifically, we focus on the case of advection of a step function by the leapfrog scheme. Numerical experiments show that the equivalent equation method fails to reproduce the oscillations generated by the scheme far from the discontinuity. Thus, we derive closed form exact and approximate solutions for the scheme that accurately predict these oscillations. We study the relationship between equivalent equation approximation and exact solution for the scheme, to determine its range of validity.