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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.09183 |
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Table of Contents:
- 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.