On the Spatial and Temporal Order of Convergence of Hyperbolic PDEs

TL;DR

This paper provides a comprehensive analysis of the global truncation error in hyperbolic PDEs, revealing how spatial and temporal discretization orders influence convergence and error behavior.

Contribution

It derives exact expressions for the global truncation error coefficients for any hyperbolic PDE and discretization method, offering new insights into convergence properties and practical implications.

Findings

Convergence order is governed by the minimum of spatial and temporal discretization orders.

Global error may increase monotonically under certain refinement scenarios.

Higher-order time-stepping methods are impractical beyond the spatial discretization order.

Abstract

In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the coefficients of the generic global truncation error. Our analysis is valid for any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method, belonging to the Method of Lines. Furthermore, we discuss the practical implications of this analysis. If we employ a stable numerical scheme and the orders of accuracy of the global solution error and the global truncation error agree, we make the following asymptotic observations: (a) the order of convergence at constant ratio of $\Delta t$ to $\Delta x$ is governed by the minimum of the orders of the spatial and temporal discretizations, and (b) convergence cannot even be guaranteed under only spatial or temporal refinement. An implication of (a) is that it is impractical to invest in a time-stepping method of order higher than the spatial discretization. In addition to (b), we demonstrate that under certain circumstances, the error can even monotonically increase with refinement only in space or only in time, and explain why this phenomenon occurs. To verify our theoretical findings, we conduct convergence studies of linear and non-linear advection equations using finite difference and finite volume spatial discretizations, and predictor-corrector and multistep time-stepping methods. Finally, we study the effect of slope limiters and monotonicity-preserving strategies on the order of accuracy.