Hyperbolic Problems Based On Natural Boundary Reduction Of Artificial Boundary Conditions

In this paper, we investigate the artificial boundary conditions and its numerical methods based on natural boundary reduction for hyperbolic initial boundary value problems.Part I, the artificial boundary condition and its numerical methods based on natural boundary reduction for hyperbolic initial boundary value problems on unbounded domains is studied. The problem is first discretized in time by the Newmark time-marching scheme, leading to a time-stepping scheme, where an exterior elliptic problem has to be solved in each time step. An artificial boundary B_r is introduced, and an exact boundary condition by the natural boundary reduction is obtained. At each time step, the variational formulation of the semidiscrete problem is derived. Existence and uniqueness of the variational solution are established. Error estimates of the fully discrete problem is performed. Stability criteria of the semi-discrete scheme is also obtained. Numerical results are presented to demonstrate the performance of our method.Part II, the artificial boundary condition and its numerical methods based on natural boundary reduction for a hyperbolic initial boundary value problem in an infinite domain with a concave angle is investigated. The governing equation is first discreted in time by Newmark method, leading to a time-stepping scheme, where an exterior elliptic problem has to be solved in each time step. Then an artificial boundary Br is introduced, an exact artificial boundary condition is obtained. The variational problem of semi-discrete problem is given, the well-posedness of the variational problem is proved, and some error estimates are presented. Finally, some numerical examples are presented to demonstrate the performance of the method.