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We propose a new method to solve the relativistic hydrodynamic equationsbased on implicit Runge-Kutta methods with a locally optimized fixed-pointiterative solver. For numerical demonstration, we implement our idea for idealhydrodynamics using the one-stage Gauss-Legendre method as an implicit method.The accuracy and computational cost of our new method are compared with thoseof explicit ones for the (1+1)-dimensional Riemann problem, as well as the(2+1)-dimensional Gubser flow and event-by-event initial conditions forheavy-ion collisions generated by TrENTo. We demonstrate that the solverconverges with only one iteration in most cases, and as a result, the implicitmethod requires a smaller computational cost than the explicit one at the sameaccuracy in these cases. By showing a relationship between the one-stageGauss-Legendre method with the iterative solver and the two-stepAdams-Bashforth method, we argue that our method benefits from both thestability of the former and the efficiency of the latter.

Efficient solver of relativistic hydrodynamics with implicit Runge-Kutta method