On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems
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arXiv
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| Format: | Preprint |
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2026
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| author | Artiano, Marco Ranocha, Hendrik |
| author_facet | Artiano, Marco Ranocha, Hendrik |
| contents | Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor's theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux differencing using summation-by-parts (SBP) operators, e.g., in discontinuous Galerkin spectral element methods (DGSEMs). The underlying two-point formulations have been extended to nonconservative systems using fluctuations by Castro et al. (2013, doi:10.1137/110845379) with follow-up generalizations to SBP methods. We propose specific forms of entropy-preserving fluctuations for nonconservative hyperbolic systems that are simple to interpret and allow an algorithmic construction of entropy-preserving methods. We analyze necessary and sufficient conditions, and obtain a full characterization of entropy-preserving three-point methods within the finite volume framework. This formulation is extended to SBP methods in multiple space dimensions on Cartesian and curvilinear meshes. Additional properties such as well-balancedness extend naturally from the underlying finite volume method to the SBP framework. We use the algorithmic construction enabled by the chosen formulation to derive several new entropy-preserving schemes for nonconservative hyperbolic systems, e.g., the compressible Euler equations of an ideal gas using the internal energy equation and a dispersive shallow-water model. Numerical experiments show the robustness and accuracy of the proposed schemes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_18978 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems Artiano, Marco Ranocha, Hendrik Numerical Analysis 65M12, 65M20, 65M70, 65M60, 65M06, 65M08 Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor's theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux differencing using summation-by-parts (SBP) operators, e.g., in discontinuous Galerkin spectral element methods (DGSEMs). The underlying two-point formulations have been extended to nonconservative systems using fluctuations by Castro et al. (2013, doi:10.1137/110845379) with follow-up generalizations to SBP methods. We propose specific forms of entropy-preserving fluctuations for nonconservative hyperbolic systems that are simple to interpret and allow an algorithmic construction of entropy-preserving methods. We analyze necessary and sufficient conditions, and obtain a full characterization of entropy-preserving three-point methods within the finite volume framework. This formulation is extended to SBP methods in multiple space dimensions on Cartesian and curvilinear meshes. Additional properties such as well-balancedness extend naturally from the underlying finite volume method to the SBP framework. We use the algorithmic construction enabled by the chosen formulation to derive several new entropy-preserving schemes for nonconservative hyperbolic systems, e.g., the compressible Euler equations of an ideal gas using the internal energy equation and a dispersive shallow-water model. Numerical experiments show the robustness and accuracy of the proposed schemes. |
| title | On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems |
| topic | Numerical Analysis 65M12, 65M20, 65M70, 65M60, 65M06, 65M08 |
| url | https://arxiv.org/abs/2603.18978 |