Restoring Convergence Order in Explicit Runge-Kutta Integration of Hyperbolic PDE with Time-Dependent Boundary Conditions

TL;DR

This paper presents a spatial correction method for explicit Runge-Kutta schemes to prevent order reduction in hyperbolic PDEs with time-dependent boundary conditions, improving convergence order.

Contribution

It introduces a boundary-adjacent operator redesign that preserves the time integrator and enables third-order convergence through algebraic boundary weight conditions.

Findings

A solvability coefficient R determines the existence of a spatial compensation mechanism.

Optimized boundary closures recover third-order convergence on classical second-order schemes.

Validation confirms improved convergence in advection and Burgers flow simulations.

Abstract

Explicit Runge-Kutta (RK) integration of hyperbolic initial-boundary value problems with time-dependent Dirichlet data often displays order reduction: the observed convergence order falls below the nominal order because the stage structure interacts with asymmetric near-boundary spatial closures. This paper develops a purely spatial remedy that preserves the time integrator while redesigning only the first two boundary-adjacent derivative operators. For an arbitrary explicit $s$-stage RK method applied to linear advection, the one-step truncation error at the boundary-adjacent nodes is shown to admit a tableau-dependent decomposition whose cancellation yields explicit algebraic conditions on the boundary weights. A solvability coefficient $R(\mathbf{b},\mathbf{c},A)$ determines whether a spatial compensation mechanism exists; the result is specialised to SSP-RK3, for which closed-form conditions are derived. Constrained differential evolution then identifies 5-point closures that, coupled to a 5th-order upwind interior stencil, recover third-order convergence from the degraded second-order behaviour of classical Taylor closures. A stability-aware variant augments the optimisation with an eigenvalue penalty, exposing the trade-off between order recovery and CFL robustness. Validation covers linear advection, manufactured-solution Burgers flow, and dimensionally split two-dimensional advection. The analysis clarifies why weak-stage-order temporal fixes do not resolve the finite-difference boundary problem, and indicates how the framework extends to non-uniform meshes.