Heat transfer is concerned with physical processes underlying the transport of thermal energy due to a temperature gradient. Our effort to develop numerical schemes for heat transfer is motivated for radiative hydrodynamics and laser fusion. Therefore, the scheme we need should have the following features:
Numerical schemes for heat transfer problems may be divided into explicit and implicit methods. An explicit scheme, i.e., forward Euler scheme, is simple, and is first order accurate, but the size of a time step is limited by a local stability condition which is normally much smaller than the required temporal accuracy. Two typical implicit schemes are backward Euler scheme and Crank-Nicolson scheme. The backward Euler scheme is first order accurate, and is useful for steady states. For the problems in radiative heat transfer and laser fusion, the temporal accuracy is important. Although Crank-Nicolson scheme is second order accurate, numerical errors do not damp in Crank-Nicolson scheme in the limit of large time steps. The numerical scheme to be presented in this talk is second-order accurate in both space and time, is stable for any size of time-steps, and numerical errors are strongly damped when a time step is large.
Implicit schemes normally involve solving a large set of algebraic equations at each time step. Exact solvers for the set of equations may not be recommended even for linear problems in two- and three-dimensions. In our scheme, we use the multigrid method to iteratively solve the set of algebraic equations. The implementation of the multigrid method in the scheme dramatically reduces the number of iteration required to reach a required accuracy. Normally, the nonlinearity is a headache in the multigrid method. The coarse grid correction normally doesn't work if the set of nonlinear equations are linearized. Since we iteratively solve the set of nonlinear equations without involving any linearization, the coarse grid correction plays an important role in our scheme. As a result, only a small number of iterations and a small amount of CPU time are needed in our scheme even for nonlinear problems.
Typically, a heat transfer problem may involve more than one kind of material. The ratio between thermal diffusivities of different material may be very large. Therefore, in principle a scheme based on Taylor expansion will not work across interfaces of different materials. Our scheme is based on the physics principle involved in the interfaces, and therefore, it works correctly and efficiently even for the problems involved in laser fusion, in which the ratio of thermal diffusivities for different materials may be as high as 10^9.
The following three figures show the convergence of three iterative approaches: Gauss-Seidel (broken line), red-black (dashed lines) and multigrid method. (solid lines).