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

Abstract

Many geophysical phenomena are characterized by properties that evolve over a wide range of scales which introduce difficulties when attempting to model these features in one computational method. We have developed a high-order finite difference method for the elastic wave equation that is able to efficiently handle varying temporal scales in a single, stand-alone framework. We apply this method to earthquake cycle models characterized by extremely long interseismic periods interspersed with abrupt, short periods of dynamic rupture. Through the use of summation-by-parts operators and weak enforcement of boundary conditions we derive a provably stable discretization. Time stepping is achieved through the implicit θ-method which allows us to take large time steps during the intermittent period between earthquakes and adapts appropriately to fully resolve rupture.

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

“…To discretize in time we use a backward-Euler adaptive-time stepping scheme [14] to numerically integrate equation (4.7) with penalty matrices given by (4.9). The initial conditions can all be specified by the initial slip velocity V(0) which we take to be V(0) =10 −7 m/s ≈10v p .…”

“…We symmetrize the matrix B = VAV −1 as before. By letting I N denote the N × N identity matrix, equation (4.7) becomes It can be shown, see [14], that the energy method applied to (4.8) and the symmetry properties of the SBP operators in combination with the penalty matrices…”

High-fidelity Sound Propagation in a Varying 3D Atmosphere

“…A well-proven HOFDM for well-posed initial boundary value problems (IBVP), is to combine summation-by-parts (SBP) operators for approximating derivatives [23,31,45], and either the simultaneous-approximation-term (SAT) method [9], or the projection method [32,[40][41][42], to impose boundary conditions (BC). Recent examples of the SBP-SAT approach can be found in [5,15,21,27,28,36,38]. The following review papers [13,49] are highly recommended for those interested in the SBP-SAT methodology.…”

On Time History Analyis With Steps Larger Than the Steps of Earthquake Records Independent From the Frequency Contents

“…Time integration is a versatile tool in analysis of transient behaviors; see [1][2][3][4][5][6][7][8][9][10]. Concentrating on structural analysis against digitized ground motion records, after discretization in space, the mathematical model can be stated as [11][12][13][14][15][16][17][18] In Eqs.…”