Alternating Direction Implicit Difference Method For Hyperbolic Equations

The study of numerical solutions for partial differential equations hold an important position in computational mathematics field. At present, the difference method is main method. In difference schemes, the explicit schemes are suitable for parallel computation but it has the limitation of stability; the implicit schemes have no stability conditions generally, but on every time step we must solve different linear systems, so it is difficult to realize parallel computation. In order to solve second order hyperbolic equations, we present operator decomposing method with add-in and variable replacement method to devise alternating direction implicit difference schemes, which have many advantages such as less computational complexity, better stability and so on. Some numerical tests prove them.In Chapter One, nonlinear elliptic differential operator can be decomposed as linear item and nonlinear item for solving a class of nonlinear second order hyperbolic equations. The former is approximated by the implicit scheme and the latter is approximated by the explicit scheme. The method can turn nonlinear problems to the similar linear system with different right-hand side at each time step, while we have similar tridiagnal matrix on left-hand, it is easy to compute and the scheme is absolutely stable. At last some numerical results are presented.In Chapter Two, variable replacement is applied to solve a class of linear second order hyperbolic equations. The advantage of this method is reduction of order: the first derivative of u to t is replaced by u, then we realize discretization to the first derivative of v to i and set up a two-layer Crank-Nicolson alternating direction implicit difference scheme. The scheme satisfied convergence at rate of second order and absolute stability. All above are proved by theory analysis. We can do with three dimensional case, then extend three dimensional one. At last some numerical results are presented.