A finite difference scheme is proposed for two-dimensional radiation hydrodynamical equations in the transport limit. The scheme is of Godunov-type, in which the set of time-averaged flux needed in the scheme is calculated through Riemann problems solved. In the scheme, flow signals are explicitly treated, while radiation signals are implicitly treated. Flow fields and radiation fields are updated simultaneously. An iterative approach is proposed to solve the set of nonlinear algebraic equations arising from the implicitness of the scheme. The sweeping method used in the scheme significantly reduces the number of iterations or computer CPU time needed. A new approach to further accelerate the convergence is proposed, which further reduces the number of iterations needed by more than one order. No matter how many cells radiation signals propagate in one time step, only an extremely small number of iterations are needed in the scheme, and each iteration costs only about 0.8 percent of computer CPU time which is needed for one time step of a second order accurate and fully explicit scheme. Two-dimensional problems are treated through a dimensionally split technique. Therefore, iterations for solving the set of algebraic equations are carried out only in each one-dimensional sweep. Through numerical examples it is shown that the scheme keeps the principle advantages of Godunov schemes for flow motion. In the time scale of flow motion numerical results are the same as those obtained from a second order accurate and fully explicit scheme. The acceleration of the convergence proposed in this paper may be directly applied to other hyperbolic systems.
Bibliographical noteFunding Information:
The work presented here has been supported by the Department of Energy through Grants DE-FG02-87ER25035 and DE-FG02-94ER25207, by the National Science Foundation through Grant ASO-9309829, by NASA through Grant USRA/5555-23/NASA, and by the University of Minnesota through its Minnesota Supercomputer Institute.
- Finite difference
- Hyperbolic system