Kristoffer Virta

Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media

We construct finite difference discretizations of the acoustic wave equation in complicated geometries and heterogeneous media. Particular emphasis is placed on the accurate treatment of interfaces at which the underlying media parameters have jump discontinuities. Discontinuous media is treated by subdividing the domain into blocks with continuous media. The equation on each block is then discretized with finite difference operators satisfying a summation-by-parts property and patched together via the simultaneous approximation term method. The energy method is used to estimate a semi-norm of the numerical solution in terms of data, showing that the discretization is stable. Numerical experiments in two and three spatial dimensions verifies the accuracy and stability properties of the schemes.

Stable Difference Methods for Block-Oriented Adaptive Grids

In this paper, we present a block-oriented scheme for adaptive mesh refinement based on summation-by-parts (SBP) finite difference methods and simultaneous-approximation-term (SAT) interface treatment. Since the order of accuracy at SBP–SAT grid interfaces is lower compared to that of the interior stencils, we strive at using the interior stencils across block-boundaries whenever possible. We devise a stable treatment of SBP-FD junction points, i.e. points where interfaces with different boundary treatment meet. This leads to stable discretizations for more flexible grid configurations within the SBP–SAT framework, with a reduced number of SBP–SAT interfaces. Both first and second derivatives are considered in the analysis. Even though the stencil order is locally reduced close to numerical interfaces and corner points, numerical simulations show that the locally reduced accuracy does not severely reduce the accuracy of the time propagated numerical solution. Moreover, we explain how to organize the grid and how to automatically adapt the mesh, aiming at problems of many variables. Examples of adaptive grids are demonstrated for the simulation of the time-dependent Schrödinger equation and for the advection equation.

Stable and high order accurate difference methods for the elastic wave equation in discontinuous media

In this paper, we develop a stable and systematic procedure for numerical treatment of elastic waves in discontinuous and layered media. We consider both planar and curved interfaces where media parameters are allowed to be discontinuous. The key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise. We discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule. Conditions at layer interfaces are imposed weakly using penalties. By deriving lower bounds of the penalty strength and constructing discrete energy estimates we prove time stability. We present numerical experiments in two space dimensions to illustrate the usefulness of the proposed method for simulations involving typical interface phenomena in elastic materials. The numerical experiments verify high order accuracy and time stability.

High Order Finite Difference Methods for the Wave Equation with Non-conforming Grid Interfaces

We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on numerical treatments of non-conforming grid interfaces and non-conforming mesh blocks. Interface conditions are imposed weakly by the simultaneous approximation term technique in combination with interface operators, which move discrete solutions between grids at an interface. In particular, we consider an interpolation approach and a projection approach with corresponding operators. A norm-compatible condition of the interface operators leads to energy stability for first order hyperbolic systems. By imposing an additional constraint on the interface operators, we derive an energy estimate of the numerical scheme for the second order wave equation. We carry out eigenvalue analyses to investigate the additional constraint and its relation to stability. In addition, a truncation error analysis is performed, and discussed in relation to convergence properties of the numerical schemes. In the numerical experiments, stability and accuracy properties of the numerical scheme are further explored, and the practical usefulness of non-conforming grid interfaces and mesh blocks is discussed in two practical examples.