Linear and Nonlinear Boundary Conditions for Wave Propagation Problems

Abstract

We discuss linear and nonlinear boundary conditions for wave propagation problems. The concepts of well-posedness and stability are discussed by considering a specific example of a boundary condition occurring in the modeling of earthquakes. That boundary condition can be formulated in a linear and nonlinear way and implemented in a characteristic and non-characteristic way. These differences are discussed and the implications and difficulties are pointed out. Numerical simulations that illustrate the theoretical discussion are presented together with an application that show that the methodology can be used for practical problems.

A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain

“…The non-conventional non-linear boundary condition in (4.4b) forces a check of wellposedness, see [25,26,33]. The energy method applied to equation (4.5) yields…”

“…Uniqueness is obtained by considering the difference problem of the form (4.5) with identical data. It was shown in [33] that this requires F (V) ≥ 0. By also observing that we give the right number of boundary conditions we can state that (4.5) in combination with (4.4b) and hence (4.4) is well posed.…”

Stable, high order accurate adaptive schemes for long time, highly intermittent geophysics problems

“…The non-conventional nonlinear boundary condition in (1b) forces a check of well-posedness, see [26,27]. Letting || · || denote the standard L 2 norm and taking the data V p = 0, the energy method applied to equation (2) yields…”