The maximal order of semidiscrete schemes for quasilinear first order partial differential equations

TL;DR

This paper establishes theoretical limits on the accuracy order of semidiscrete schemes for quasilinear first order PDEs, showing that the maximal order is bounded by the number of points used, with stability considerations further reducing this bound.

Contribution

It provides a rigorous proof of the maximal achievable order for semidiscrete schemes and links stability of the fully discrete scheme to the order bound.

Findings

Maximal order of semidiscrete schemes is 2r for (2r+1)-point schemes.

Stable fully discrete schemes have an upper order bound of 2r-1.

The bounds are attained by many schemes and equations.

Abstract

We prove that a semidiscrete $(2r+1)$-point scheme for quasilinear first order PDE cannot attain an order higher than $2r$. Moreover, if the forward Euler fully discrete scheme obtained from the linearization about any constant state of the semidiscrete scheme is stable, then the upper bound for the order of the scheme is $2r-1$. This bound is attained for a wide range of schemes and equations.