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High-order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-type Equation of State
Linfeng Xu, Shengrong Ding, Kailiang Wu
TL;DR
Existing RHD entropy-stable schemes were restricted to the ideal EOS, which is often inaccurate for relativistic flows. This paper develops high-order schemes for general Synge-type EOSs by establishing a convex entropy structure and constructing unified EC/ES discretizations. Numerical examples validate their accuracy and effectiveness across special EOSs, while exact EC-flux construction can require EOS-specific evaluation of an integral.
Problem
Existing entropy-stable RHD schemes were limited to the ideal EOS, which is often a poor approximation for relativistic flows.
Method
The paper establishes a strictly convex entropy pair and entropy variables, constructs unified two-point EC fluxes, and adds ENO/WENO-based dissipation for arbitrarily high-order ES schemes.
Results
Numerical examples demonstrate the accuracy and effectiveness of the proposed schemes for RHD with four special EOSs.
Takeaways & Limitations
The framework extends high-order entropy-stable RHD discretization from the ideal EOS to a general Synge-type EOS class.
Takeaways & Limitations
For complicated EOS functions, exact evaluation of the integral required by the EC property may be difficult, motivating alternative EOS-specific formulations.
Abstract
from arXiv · showhide
All the existing entropy stable (ES) schemes for relativistic hydrodynamics (RHD) in the literature were restricted to the ideal equation of state (EOS), which however is often a poor approximation for most relativistic flows due to its inconsistency with the relativistic kinetic theory. This paper develops high-order ES finite difference schemes for RHD with general Synge-type EOS, which encompasses a range of special EOSs. We first establish an entropy pair for the RHD equations with general Synge-type EOS in any space dimensions. We rigorously prove that the found entropy function is strictly convex and derive the associated entropy variables, laying the foundation for designing entropy conservative (EC) and ES schemes. Due to relativistic effects, one cannot explicitly express primitive variables, fluxes, and entropy variables in terms of conservative variables. Consequently, this highly complicates the analysis of the entropy structure of the RHD equations, the investigation of entropy convexity, and the construction of EC numerical fluxes. By using a suitable set of parameter variables, we construct novel two-point EC fluxes in a unified form for general Synge-type EOS. We obtain high-order EC schemes through linear combinations of the two-point EC fluxes. Arbitrarily high-order accurate ES schemes are achieved by incorporating dissipation terms into the EC schemes, based on (weighted) essentially non-oscillatory reconstructions. Additionally, we derive the general dissipation matrix for general Synge-type EOS based on the scaled eigenvectors of the RHD system. We also define a suitable average of the dissipation matrix at the cell interfaces to ensure that the resulting ES schemes can resolve stationary contact discontinuities accurately. Several numerical examples are provided to validate the accuracy and effectiveness of our schemes for RHD with four special EOSs.
1 Introduction
The paper extends entropy-stable numerical methods for relativistic hydrodynamics beyond the ideal equation of state to general Synge-type EOSs. It establishes the entropy structure and develops high-order entropy-conservative and entropy-stable schemes for this broader class.
- General Synge-type EOSs include the ideal, RC, IP, and TM EOSs while requiring only relatively mild assumptions compatible with commonly used models.
- Existing entropy-conservative and entropy-stable RHD schemes were limited to the ideal EOS, despite its poor approximation for many relativistic astrophysical flows.
- The nonlinear coupling prevents explicit expressions of primitive variables, fluxes, and entropy variables in conservative variables, complicating entropy analysis and EC-flux construction.
- The paper establishes a strictly convex entropy pair and associated entropy variables for general Synge-type EOSs in any space dimension.
- Unified two-point EC fluxes support high-order EC schemes, while ENO/WENO-based dissipation yields arbitrarily high-order ES schemes with exact stationary-contact resolution.
2 Entropy analysis for RHD equations
This section constructs and analyzes an entropy pair for RHD with general Synge-type EOSs. The entropy is proved strictly convex under the relativistic causality condition, and its entropy variables and potential fluxes are derived.
- 2.1 Entropy pair: An entropy pair is established for the RHD system with general Synge-type EOSs, including the four listed special EOSs.
- 2.1 Entropy pair: The entropy identity is verified by expressing entropy, fluxes, and conservative variables through primitive variables and applying the chain rule.
- 2.2 Convexity of entropy function: Convexity is proved by showing positive definiteness of the entropy Hessian through matrix transformations and the causality condition.
- 2.3 Entropy variables: The associated entropy variables and potential fluxes are derived explicitly and provide the quantities needed to construct entropy-stable schemes.
3 1D entropy stable schemes
The 1D construction develops unified two-point entropy-conservative fluxes for general Synge-type EOS and extends them to high-order entropy-conservative and entropy-stable schemes.
- 3.1 Two-point entropy conservative flux: A unified two-point entropy-conservative flux is derived for the 1D RHD system with general Synge-type EOS using parameter variables z = (z1, z2, z3).The formulation applies across the general EOS class, while the auxiliary quantity E depends on the adopted EOS.
- 3.1 Two-point entropy conservative flux: The flux construction reduces the entropy-conservation condition to relations involving jumps and arithmetic averages of the chosen parameter variables and entropy variables.The proof expresses entropy variables and the potential flux in the parameter variables, then solves the resulting coefficient conditions.
- 3.1 Two-point entropy conservative flux: The quantity E requires an EOS-dependent integral that may be difficult to express explicitly for complicated internal-energy functions.Exact evaluation of this integral is required to achieve the entropy-conservative property exactly.
- 3.1 Two-point entropy conservative flux: For ID-EOS, RC-EOS, IP-EOS, and TM-EOS, explicit formulas for E are provided within the same unified flux form.These four EOSs are special cases of the general Synge-type class and permit explicit evaluation of the integral-dependent quantity.
- 3.2.1 Second-order entropy conservative schemes: The second-order scheme obtained from the two-point entropy-conservative flux satisfies a discrete entropy equality and is therefore entropy conservative.The numerical entropy flux is consistent with the physical entropy flux.
- 3.2 Entropy conservative schemes: Linear combinations of the two-point flux produce higher-order entropy-conservative schemes, while ENO or WENO-based dissipation yields high-order entropy-stable schemes.The resulting entropy-stable fluxes satisfy discrete entropy inequalities; the ENO- and WENO-based constructions are explicitly identified as entropy stable.
4 2D entropy stable schemes
The 2D construction extends the unified entropy-conservative flux and dissipation-matrix framework dimension by dimension for general Synge-type EOS.
- 4 2D entropy stable schemes: The 2D entropy-conservative and entropy-stable schemes are constructed dimension by dimension, with two-point fluxes and dissipation matrices as key ingredients.The derivation is analogous to the 1D case and is presented separately for each spatial direction.
- 4.1 Two-point entropy conservative flux: A unified 2D two-point entropy-conservative flux is derived using parameter variables z = (z1, z2, z3, z4) and the entropy potential.The construction rewrites entropy-variable and potential-flux jumps in these variables before solving for the flux components.
- 4.2 Dissipation matrix: The 2D dissipation matrices are formed from suitably scaled right eigenvectors of the directional RHD Jacobians.Separate matrices correspond to the x1- and x2-directions, with the scaled eigenvector matrices satisfying the required entropy-variable relation.
- 4.2 Dissipation matrix: The multidimensional eigenvector verification is more complicated because the Lorentz factor couples the two velocity components.The derivation uses rotational symmetry between the coordinate directions to relate the directional matrices.
- 4.2 Dissipation matrix: Interface averages for the dissipation matrices are selected to accurately resolve stationary contact discontinuities.The averaged enthalpy-like quantity is chosen appropriately, while other averages follow the 1D construction.
- 4 2D entropy stable schemes: The resulting semidiscrete schemes can be advanced with high-order Runge–Kutta methods, including SSP-RK3.The paper notes that rigorous fully discrete entropy stability is not yet available for the SSP-RK3 coupling.
5 Numerical experiments
Numerical experiments across one- and two-dimensional RHD problems and four special EOSs confirm the schemes' expected accuracy, entropy behavior, and ability to resolve discontinuous wave structures.
- Accuracy tests: The schemes achieve their expected convergence orders for smooth one- and two-dimensional tests.The reported errors and convergence rates cover EC6, ES5, EC4, and ES4 across multiple EOSs and grid resolutions.
- Entropy behavior: EC6 and EC4 keep discrete total entropy nearly unchanged, whereas ES5 and ES4 decrease it through numerical dissipation, with smaller decay on finer grids.This behavior is reported for the one-dimensional accuracy tests and reflects the intended distinction between entropy-conservative and entropy-stable schemes.
- Entropy-stable comparisons: The non-ES5 scheme produces spurious oscillations and violates the discrete entropy inequality, while EC6 and ES5 better resolve the solution structure.The comparison uses SSP-RK3 and 200 uniform grids.
- One-dimensional examples: ES5 resolves one-dimensional shocks, contacts, rarefactions, and blast-wave structures while maintaining decreasing discrete total entropy.These results include tests with RC-EOS, IP-EOS, and TM-EOS and comparisons with reference solutions.
- Time discretization: SSP-RK3 and RRK3 produce very similar numerical results for both smooth and discontinuous problems.The subsequent experiments therefore use SSP-RK3 to save space.
- Two-dimensional examples: In two dimensions, ES5 captures complex relativistic waves, shock interactions, bubble dynamics, and mushroom-cloud formation for RC-EOS, IP-EOS, and TM-EOS.The reported simulations use 400×400 or 650×180 uniform grids and show the relevant wave interactions.
6 Conclusions
The paper develops high-order entropy stable schemes for RHD with general Synge-type equations of state, addressing limitations of ideal-EOS methods. It establishes the entropy structure, constructs unified entropy-conservative fluxes and dissipation mechanisms, and validates the schemes numerically.
- The paper makes a first attempt to develop high-order ES finite difference schemes for RHD with general Synge-type EOS covering more accurate special EOSs.
- It discovers an entropy pair, proves strict convexity of the entropy function, and derives the associated entropy variables for general Synge-type EOS.
- Using suitable parameter variables, it constructs explicit two-point EC fluxes in unified form, then obtains higher-order EC schemes through linear combinations.
- Adding ENO or WENO-based dissipation produces arbitrarily high-order accurate ES schemes, supported by a general dissipation matrix based on scaled RHD eigenvectors.
- Numerical examples with several special EOSs demonstrate the accuracy and effectiveness of the proposed schemes.
- The results support further development of ES discontinuous Galerkin or finite volume schemes and EC/ES schemes for relativistic MHD with Synge-type EOS.