We introduce an energy stable, high-order-accurate finite difference approximation of the dynamic, pure bending Kirchhoff plate equations for complex geometries and spatially variable properties. We utilize the summation-by-parts (SBP) framework to discretize the biharmonic operator with variable coefficients, with attention given to free and clamped boundary conditions and corner conditions. Energy conservation is established by combining SBP boundary closures with weak enforcement of the boundary and interface conditions using a penalty (simultaneous approximation term, SAT) technique. Then we couple the plate equations to the shallow water equations to study flexural-gravity wave propagation, and prove that the semi-discrete system of equations is self-adjoint. We demonstrate the stability and accuracy properties of the method on curvilinear multiblock grids using the method of manufactured solutions. The method, which we provide in an open-source code, is then used to model ocean wave interactions with the Thwaites Glacier and Pine Island Ice Shelf in the Amundsen Sea off the coast of West Antarctica.

We consider energy stable summation by parts finite difference methods (SBP-FD) for the homogeneous and piecewise homogeneous dynamic beam equation (DBE). Previously the constant coefficient problem has been solved with SBP-FD together with penalty terms (SBP-SAT) to impose boundary conditions. In this work we revisit this problem and compare SBP-SAT to the projection method (SBP-P). We also consider the DBE with discontinuous coefficients and present novel SBP-SAT, SBP-P and hybrid SBP-SAT-P discretizations for imposing interface conditions. Numerical experiments show that all methods considered are similar in terms of accuracy, but that SBP-P can be more computationally efficient (less restrictive time step requirement for explicit time integration methods) for both the constant and piecewise constant coefficient problems.

A high-order finite difference discretisation of the Schrodinger equation on domains that deform in time is presented. For the domain deformation, a time-dependent coordinate transformation is used. The utilisation of summation-by-parts finite difference operators, combined with boundary conditions imposed weakly by a penalty technique, leads to a provably stable scheme. The convergence properties of the scheme are verified by numerical computations. The capabilities of the numerical scheme are demonstrated by simulations of Berry phases in deforming quantum billiards. (C) 2021 The Author(s). Published by Elsevier Inc.

Seismic air guns are broadband sources that generate acoustic waves at many frequencies. The low-frequency waves can be used for imaging, whereas the high-frequency waves are attenuated and/or scattered before they can reflect from targets of interest in the subsurface. It is desirable to reduce the amplitude of the high-frequency acoustic waves because they are thought to be disruptive, and potentially damaging, to marine life and are not useful for geophysical purposes. The high-frequency acoustic waves are primarily associated with the initial expansion of the air-gun bubble and associated peak in the acoustic pressure time series, which is commonly referred to as the source signature of the air gun. We have developed a quasi-1D model of a seismic air gun coupled to a spherical bubble that accounts for gas dynamics and spatially variable depressurization inside the firing chamber to investigate controls on the initial peak of the source signature. The model is validated against data collected during field tests in Lake Seneca, New York. Simulations and field data show that the initial peak is primarily dependent on the operating pressure. A lower gun pressure results in a smaller peak amplitude and a slower rise time. The slope, the amplitude of the initial peak divided by the rise time, is used as a proxy for environmental impact and can decrease by as much as 50% when the air-gun pressure is reduced from 2000 to 1000 psi. The low frequencies are controlled by the total discharged mass. which is dependent upon the gun volume and pressure. Decreasing the operating pressure while simultaneously increasing the gun volume will reduce the high frequencies while maintaining the desirable low-frequency signals.

We study non-conforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the non-conforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.

Energy-stable numerical methods for initial boundary value problems (IBVP) often rely on a summation-by-parts (SBP) property. The SBP property ensures that the discrete differential operators mimic the integration-by-parts formula for their continuous counterparts. Examples of discrete operators with the SBP property include some finite difference (FD)[1], spectral collocation [2], and finite volume methods [3]. SBP FD operators are routinely combined with the simultaneous approximation term (SAT) method to impose boundary conditions. The combination of SBP operators and the SAT technique allow us to construct numerical methods that exactly mimic the energy growth of the underlying IBVP. The SBP-SAT method is a robust numerical technique for multi-physics problems and hybrid numerical methods, combining different SBP discretisations. Another method of imposing boundary conditions in the framework of SBP, that has recently gained attention is the the projection method [4]. In [5] a multi-physics model is constructed for simulation of acoustic and flexural gravity waves. Such waves arise when studying ocean wave interactions with floating ice shelves, sea ice, and floating structures. Particular emphasis is on a well-posed interface treatment of the fluid-ice coupling. To ensure numerical stability and efficiency, a high-order accurate SBP-SAT method is employed to impose the boundary and the multi-physics interface conditions. In the present study we will show how the SBP-SAT (or Projection) method can be employed to ensure numerical stability and efficiency in the framework of well-posed multi-physics IBVP. As a proof of concept we will show results from the analysis and convergence study of acoustic and flexural gravity waves in compressible, inviscid fluids partially covered by a thin elastic layer [5]. We will also show some recent results concerning a coupled (multi-physics) model of a seismic air gun, solved with the SBP-SAT method. This is an extension of the work presented in [6].

We introduce a provably stable, high-order-accurate finite difference method for simulation of acoustic and flexural gravity waves in compressible, inviscid fluids partially covered by a thin elastic layer. Such waves arise when studying ocean wave interactions with floating ice shelves, sea ice, and floating structures. Particular emphasis is on a well-posed interface treatment of the fluid-ice coupling. To ensure numerical stability and efficiency, finite difference approximations based on the summation-by-parts (SBP) framework are combined with a penalty technique (simultaneous approximation term, SAT) to impose the boundary and interface conditions. The resulting SBP-SAT approximations are time integrated with an unconditionally stable finite difference method. Numerical simulations in 2D corroborate the predicted efficiency and stability behaviors. The method can be used in its current form to study transmission of ocean waves and tsunamis through ice shelves, and upon coupling to an elastic half-space beneath the ice and water, to study ice motions associated with long-period seismic surface waves.

A stable and high-order accurate upwind finite difference discretization of the 3D linearized Euler equations is presented. The discretization allows point sources, a varying atmosphere and curved topography. The advective terms are discretized using recently published upwind summation-by-parts (SBP) operators and the boundary conditions are imposed using a penalty technique. The resulting discretization leads to an explicit ODE system. The accuracy and stability properties are verified for a linear hyperbolic problems in 1D, and for the 3D linearized Euler equations. The usage of upwind SBP operators leads to robust and accurate numerical approximations in the presence of point sources and naturally avoids the onset of spurious oscillations.

Finite difference methods are commonly used for spatial discretization of partial differential equations (PDEs). They provide a simple and potentially highly efficient means of approximating solutions of both time dependent and time independent problems. If designed and implemented correctly they can also be very robust and make efficient use of computer resources. Moreover, they can also provide high order schemes without putting severe constraints on time steps when integrating in time. Traditionally, finite-difference methods have been designed, analyzed and implemented on a problem by problem basis. This has resulted in specific codes for specific equations and specific boundary conditions. While this shortens implementation time for the individual developer in the short term there has been and will be a lot of duplicated effort designing and optimizing software interfaces. A potential explanation for this is that for a long time there were no tools for easily tackling new problems and equations. As a result there seems to be a lack of a general high-performance software library for large-scale finite-difference computations. During the last 20 years, research has shown that the summation by parts framework (SBP) for finite-difference methods combined with the weak imposition of boundary conditions using the simultaneous approximation term technique (SAT) can provide such tools. The framework provides mathematical tools that allow deriving accurate and stable finite-difference schemes for many combinations of PDE and boundary conditions. A few examples can be found in [11, 4]. While general frameworks such as Matlab or SciPy have successfully been used in many cases to prove the applicability of the schemes, they lack the ability to scale to large problems. Once large-scale simulations are to be performed, where performance is vital, implementations in low-level languages

High-order accurate finite difference schemes are derived for a non-linear soliton model of nerve signal propagation in axons. Two types of well-posed boundary conditions are analysed. The boundary closures are based on the summation-by-parts (SBP) framework and the boundary conditions are imposed using a penalty (SAT) technique, to guarantee linear stability. The resulting SBP–SAT approximation is time-integrated with an explicit finite difference method. The accuracy and stability properties of the newly derived finite difference approximations are demonstrated for an analytic soliton solution.

We introduce a coarse-graining approach to study the movement of a Domain Wall (DW) under the influence of a spin polarized current, in the framework of atomistic spin dynamics. An increase in performance of up to 35% is obtained. We show the dependence of the method on both exchange range and temperature effects.

A High-order accurate finite difference scheme is derived for a non-linear soliton model of nerve signal propagation in axons. Boundary conditions yielding well-posed problems are suggested and inc ...