Efficient Solver of Relativistic Hydrodynamics with an Implicit Runge–Kutta Method

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)D Riemann problem, as well as the (2+1)D 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 Δt. 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.