On the implementation of Absorbing Boundary Conditions in a Finite Difference code with conventional grid

Abstract

Modern numerical experiments for the solution of the direct problem in Seismology (i.e., the elasto–dynamic problem for fault surfaces) require the use of advanced numerical algorithms, capable of capturing all the essential features of the physical problem and to properly resolve the characteristic temporal and spatial lengths. In realistic dynamic models the fault surfaces have dimensions of several kilometer, in both strike and dip directions (e.g., Bizzarri et al., 2009, among many others). The required resolution of the problem typically requires the adoption of spatial sampling of several meter and time steps of the order of fractions of millisecond. As a consequence, this results in numerical experiments with algebraic equations discretized over hundreds of mega–nodes (n x 108 nodes). In turn, in order to obtain results in affordable human–times, this requires the exploitation of symmetry conditions (see for instance Bizzarri, 2009 for further details) and the use of several code optimizations, as well as an efficient parallel programming. In addition to the grid dispersion phenomenon, intrinsically present in every numerical algorithm, another problem can affect the obtained solutions: the spurious reflections of signals originated from the boundaries of the computational domain. These reflections might introduce numerical artifacts into the computed solutions and constructively interfere, finally causing problematic oscillations. One way to solve this problem is to arbitrarily enlarge the size of the computational domain — ideally approaching the unbounded (with the exclusion of the free surface) medium — in order to delay the back propagating fronts originating from the model boundaries. This solution is theoretically optimal, but technically unpractical, since the size of the model can easily become larger than the available computational resources. A second possibility to assess the problem is to introduce some ad hoc conditions at (or near to) the boundaries of the computational domain, in order to cause the back propagating waves to be adequately small (ideally null) from a numerical point of view. In this study we present different numerical algorithms consisting in Absorbing Boundary Conditions (ABCs thereinafter) that can be efficiently used to reduce the boundary effects (i.e., the waves originated by a seismogenic fault of finite extension reflected back into the model by the boundaries of the computational domain). We also indicate how they can be proficiently implemented in a Finite Difference, conventional grid numerical FORTRAN code.