Difference Schemes for Multi-dimensional Problems

In this chapter, the extension of one-dimensional explicit schemes to multi-dimensional problems, which has stricter stability restrictions, is discussed. While the stable implicit schemes in multi-dimensional problems are not tridiagonal, which limits the advantage of fast solving. To realize the fast calculation, alternate direction method and an operator split technique are introduced in the rest of this chapter. The alternating direction method is to divide the time step into two halves and use implicit and explicit formats in different spatial directions respectively, thus transforming the original five-diagonal matrix equation into two tridiagonal matrix equations, which can be solved by fast algorithms. While the basic idea of splitting operator runs through almost all general numerical methods in fluid mechanics, which includes time-splitting, direction-splitting, physical phenomenon splitting, and so on.