Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

In this article,discrete variants of several results from vector calculus are studied for clas-sical finite difference summation by parts operators in two and three space dimensions.It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations,which are characterised explicitly.This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition.Nevertheless,iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are pro-posed and applied successfully.In numerical experiments,the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first-order partial differential equations.Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics,applications to the discrete analysis of magnetohydrodynamic(MHD) wave modes are presented and discussed.

Abel’s lemma on summation by parts and partial q-series transformations

The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q-series are established.

INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY 2F1(1/2)-SUMS

The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type,including a couple of beautiful expressions forπand the Catalan constant discovered by Guillera(2008).

Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

We develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two-and three-dimensional spatial domains.In this method,waves are slowed down and dissipated in sponge layers near the far-field boundaries.Mathematically,this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain,where the elastic wave equation is solved numerically on a regular grid.To damp out waves that become poorly resolved because of the coordinate mapping,a high order artificial dissipation operator is added in layers near the boundaries of the computational domain.We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy,which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain.Our spatial discretization is based on a fourth order accurate finite difference method,which satisfies the principle of summation by parts.We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries.Therefore,the coefficients in the finite difference stencils need only be boundary modified near the free surface.This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains.Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer.The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem,where fourth order accuracy is observed with a sixth order artificial dissipation.We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.

Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

We develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation.The material model is a super-imposition of N standard linear solid mechanisms,which commonly is used in seismology to model a material with constant quality factor Q.The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables,making it significantly more memory efficient than the commonly used first order velocitystress formulation.The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation[SIAM J.Numer.Anal.,45(2007),pp.1902–1936].Our main result is a proof that the proposed discretization is energy stable,even in the case of variable material properties.The proof relies on the summation-by-parts property of the discretization.The newscheme is implemented with grid refinement with hanging nodes on the interface.Numerical experiments verify the accuracy and stability of the new scheme.Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy.We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

Stable Grid Refinement and Singular Source Discretization for Seismic Wave Simulations

An energy conserving discretization of the elastic wave equation in second order formulation is developed for a composite grid,consisting of a set of structured rectangular component grids with hanging nodes on the grid refinement interface.Previously developed summation-by-parts properties are generalized to devise a stable second order accurate coupling of the solution across mesh refinement interfaces.The discretization of singular source terms of point force and point moment tensor type are also studied.Based on enforcing discrete moment conditions that mimic properties of the Dirac distribution and its gradient,previous single grid formulas are generalized to work in the vicinity of grid refinement interfaces.These source discretization formulas are shown to give second order accuracy in the solution,with the error being essentially independent of the distance between the source and the grid refinement boundary.Several numerical examples are given to illustrate the properties of the proposed method.

Stability of Finite Difference Discretizations of Multi-Physics Interface Conditions

We considermulti-physics computationswhere theNavier-Stokes equations of compressible fluid flow on some parts of the computational domain are coupled to the equations of elasticity on other parts of the computational domain.The different subdomains are separated by well-defined interfaces.We consider time accurate computations resolving all time scales.For such computations,explicit time stepping is very efficient.We address the issue of discrete interface conditions between the two domains of different physics that do not lead to instability,or to a significant reduction of the stable time step size.Finding such interface conditions is non-trivial.We discretize the problem with high order centered difference approximations with summation by parts boundary closure.We derive L2 stable interface conditions for the linearized one dimensional discretized problem.Furthermore,we generalize the interface conditions to the full non-linear equations and numerically demonstrate their stable and accurate performance on a simple model problem.The energy stable interface conditions derived here through symmetrization of the equations contain the interface conditions derived through normal mode analysis by Banks and Sj¨ogreen in[8]as a special case.

New Proof for Some Terminating Hypergeometric Series Identities

The Abel＇s lemma on summation by parts is employed to evaluate terminating hypergeometric series. Several summation formulae are reviewed and some new identities are established.