Efficient solver of relativistic hydrodynamics with implicit Runge-Kutta method

TL;DR

This paper introduces an implicit Runge-Kutta based solver for relativistic hydrodynamics that achieves high accuracy with reduced computational cost by using a locally optimized fixed-point iterative method, demonstrated on complex flow problems.

Contribution

The paper presents a novel implicit Runge-Kutta solver with a fixed-point iterative approach for relativistic hydrodynamics, combining stability and efficiency advantages.

Findings

Converges with only one iteration in most cases.

Requires less computational cost than explicit methods at the same accuracy.

Effective for complex flow problems like Gubser flow and heavy-ion collision initial conditions.

Abstract

We propose a new method to solve the relativistic hydrodynamic equations based on implicit Runge-Kutta methods with a locally optimized fixed-point iterative solver. For numerical demonstration, we implement our idea for ideal hydrodynamics using the one-stage Gauss-Legendre method as an implicit method. The accuracy and computational cost of our new method are compared with those of explicit ones for the (1+1)-dimensional Riemann problem, as well as the (2+1)-dimensional Gubser flow and event-by-event initial conditions for heavy-ion collisions generated by TrENTo. We demonstrate that the solver converges with only one iteration in most cases, and as a result, the implicit method requires a smaller computational cost than the explicit one at the same accuracy in these cases, while it may not converge with an unrealistically large $\Delta t$. By showing a relationship between the one-stage Gauss-Legendre method with the iterative solver and the two-step Adams-Bashforth method, we argue that our method benefits from both the stability of the former and the efficiency of the latter.