Energy-conserving 3D elastic wave simulation with finite difference discretization on staggered grids with nonconforming interfaces

In this work, we describe an approach to stably simulate the 3D isotropic elastic wave propagation using finite difference discretization on staggered grids with nonconforming interfaces. Specifically, we consider simulation domains composed of layers of uniform grids with different grid spacings, separated by planar interfaces. This discretization setting is motivated by the observation that wave speeds of earth media tend to increase with depth due to sedimentation and consolidation processes. We demonstrate that the layer-wise finite difference discretization approach has the potential to significantly reduce the simulation cost, compared to its counterpart that uses holistically uniform grids. Such discretizations are enabled by summation-by-parts finite difference operators, which are standard finite difference operators with special adaptations near boundaries or interfaces, and simultaneous approximation terms, which are penalty terms appended to the discretized system to weakly impose boundary or interface conditions. Combined with specially designed interpolation operators, the discretized system is shown to preserve the energy-conserving property of the continuous elastic wave equation, and a fortiori ensure the stability of the simulation. Numerical examples are presented to corroborate these analytical developments

On discretely entropy conservative and entropy stable discontinuous Galerkin methods

High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions

Modelling Seismic Wave Propagation for Geophysical Imaging

International audienceThe Earth is an heterogeneous complex media from the mineral composition scale (10−6m) to the global scale ( 106m). The reconstruction of its structure is a quite challenging problem because sampling methodologies are mainly indirect as potential methods (Günther et al., 2006; Rücker et al., 2006), diffusive methods (Cognon, 1971; Druskin & Knizhnerman, 1988; Goldman & Stover, 1983; Hohmann, 1988; Kuo & Cho, 1980; Oristaglio & Hohmann, 1984) or propagation methods (Alterman & Karal, 1968; Bolt & Smith, 1976; Dablain, 1986; Kelly et al., 1976; Levander, 1988; Marfurt, 1984; Virieux, 1986). Seismic waves belong to the last category. We shall concentrate in this chapter on the forward problem which will be at the heart of any inverse problem for imaging the Earth. The forward problem is dedicated to the estimation of seismic wavefields when one knows the medium properties while the inverse problem is devoted to the estimation of medium properties from recorded seismic wavefields

Low-Dissipation Simulation Methods and Models for Turbulent Subsonic Flow

The simulation of turbulent flows by means of computational fluid dynamics is highly challenging. The costs of an accurate direct numerical simulation (DNS) are usually too high, and engineers typically resort to cheaper coarse-grained models of the flow, such as large-eddy simulation (LES). To be suitable for the computation of turbulence, methods should not numerically dissipate the turbulent flow structures. Therefore, energy-conserving discretizations are investigated, which do not dissipate energy and are inherently stable because the discrete convective terms cannot spuriously generate kinetic energy. They have been known for incompressible flow, but the development of such methods for compressible flow is more recent. This paper will focus on the latter: LES and DNS for turbulent subsonic flow. A new theoretical framework for the analysis of energy conservation in compressible flow is proposed, in a mathematical notation of square-root variables, inner products, and differential operator symmetries. As a result, the discrete equations exactly conserve not only the primary variables (mass, momentum and energy), but also the convective terms preserve (secondary) discrete kinetic and internal energy. Numerical experiments confirm that simulations are stable without the addition of artificial dissipation. Next, minimum-dissipation eddy-viscosity models are reviewed, which try to minimize the dissipation needed for preventing sub-grid scales from polluting the numerical solution. A new model suitable for anisotropic grids is proposed: the anisotropic minimum-dissipation model. This model appropriately switches off for laminar and transitional flow, and is consistent with the exact sub-filter tensor on anisotropic grids. The methods and models are first assessed on several academic test cases: channel flow, homogeneous decaying turbulence and the temporal mixing layer. As a practical application, accurate simulations of the transitional flow over a delta wing have been performed

On the energy stability of high-order finite volume schemes for initial-boundary value problems

We examine the energy stability of high-order finite volume schemes approximating linear hyperbolic initial- boundary value problems. In particular, we consider schemes obtained by the k-exact method and the spectral volume method using the central numerical flux. To determine the stability of the schemes we use the energy method, and investigate the resulting terms. Finally, we compute numerical results verifying the accuracy of the schemes.Masteroppgave i anvendt og beregningsorientert matematikkMAB399MAMN-MA