[Paper Review] Two-stage fourth-order accurate time discretizations for 1D and 2D special relativistic hydrodynamics

Yuhuan Yuan, Huazhong Tang|arXiv (Cornell University)|Dec 15, 2017

Advanced Numerical Methods in Computational Mathematics33 references7 citations

TL;DR

This paper proposes a two-stage fourth-order accurate time discretization for 1D and 2D special relativistic hydrodynamics, leveraging the direct Eulerian generalized Riemann problem (GRP) method and analytical resolution of local quasi-1D GRP states. The scheme achieves high-order accuracy and robustness in capturing shocks and complex wave structures, validated through numerical experiments on standard Riemann problems and smooth flow tests.

ABSTRACT

This paper studies the two-stage fourth-order accurate time discretization \cite{LI-DU:2016} and applies it to special relativistic hydrodynamical equations. It is shown that new two-stage fourth-order accurate time discretizations can be proposed. With the aid of the direct Eulerian GRP (generalized Riemann problem) methods \cite{Yang-He-Tang:2011,Yang-Tang:2012} and the analytical resolution of the local "quasi 1D" GRP, the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations. Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.

Motivation & Objective

• To develop higher-order accurate numerical schemes for special relativistic hydrodynamics (RHD) that maintain robustness and accuracy in shock-capturing.
• To extend the two-stage fourth-order time discretization framework to 1D and 2D RHD equations, overcoming the computational cost of traditional Runge-Kutta methods.
• To implement the scheme using the direct Eulerian GRP method with analytical resolution of local quasi-1D GRP states for improved efficiency and accuracy.
• To validate the scheme’s performance through comprehensive numerical experiments on Riemann problems and smooth flows.

Proposed method

• Adopts a two-stage fourth-order time discretization derived from the GRP flux function, avoiding the need for full-stage Riemann solver calls.
• Employs the direct Eulerian GRP method to compute time derivatives of fluxes up to second order via analytical resolution of local Riemann problems.
• Uses the analytical solution of the local quasi-1D GRP to compute high-order temporal derivatives of the conservative variables, enabling fourth-order accuracy in time.
• Applies the scheme to the 1D and 2D special relativistic Euler equations in conservation form, with a $̳$-law equation of state.
• Derives evolution equations for time derivatives of primitive variables using characteristic analysis and Riemann invariants, particularly for the acoustic and shock cases.
• Implements the scheme with a consistent treatment of the Lorentz factor and relativistic fluxes, ensuring physical consistency and stability.

Experimental results

Research questions

• RQ1 Can a two-stage fourth-order time discretization be effectively adapted to the 1D and 2D special relativistic hydrodynamics equations?
• RQ2 How does the proposed scheme compare in accuracy and efficiency to traditional Runge-Kutta methods for RHD?
• RQ3 Can the direct Eulerian GRP method with analytical GRP resolution enable high-order temporal accuracy without excessive computational cost?
• RQ4 What is the robustness of the scheme in resolving strong shocks and rarefaction waves in relativistic flows?
• RQ5 How does the scheme perform on standard 1D and 2D Riemann problems and smooth flow benchmarks?

Key findings

• The proposed two-stage fourth-order time discretization achieves fourth-order accuracy in time for both 1D and 2D special relativistic hydrodynamics, as confirmed by convergence studies.
• The scheme successfully resolves complex wave structures, including shocks and rarefactions, with high resolution and minimal oscillations.
• Numerical experiments show that the method maintains robustness even in extreme relativistic regimes, such as high Lorentz factors and strong shocks.
• The computational cost is significantly reduced compared to standard four-stage Runge-Kutta methods, as each stage avoids full Riemann solver calls.
• The scheme demonstrates optimal convergence rates in L1 and L∞ norms for smooth solutions, confirming the theoretical order of accuracy.
• The analytical resolution of the local quasi-1D GRP enables accurate and efficient computation of time derivatives up to second order, forming the core of the high-order accuracy.