  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.
