Simultaneous Approximation Term Research Articles

Upwind Summation-by-parts Finite Differences: error Estimates and WENO methodology

High order upwind summation-by-parts finite difference operators have recently been developed. When combined with the simultaneous approximation term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities.

Multiresolution approximation for shallow water equations using summation-by-parts finite differences

Abstract We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.

A stable scheme of the Curvilinear Shallow Water Equations with no-penetration and far-field boundary conditions

This paper presents a stable and highly accurate numerical tool for computing river flows in urban areas, which is a first step towards a numerical tool for flood predictions. We start with the (linearized) well-posedness analysis by Ghader and Nordström (2014), where far-field boundary conditions were proposed and extend their analysis to include wall boundaries. Specifically, we employed high-order Summation-by-parts (SBP) finite-difference operators to construct a scheme for the Shallow Water Equations. We also developed a stable SBP scheme with Simultaneous Approximation Terms that impose far-field and wall boundaries. Finally, we extended the schemes and their stability proofs to non-Cartesian domains. To demonstrate the strength of the schemes, we performed computations for problems with exact solutions to obtain second, third, and fourth (2, 3, 4) convergence rates. Finally, we applied the 4th-order scheme to steady river channels, the canal (or flood-control channel simulations), and dam-break problems. The results show that the imposition of the boundary conditions is stable, and the far-field boundaries cause no visible reflections at the boundaries.

A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations

In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner. We first derive continuous energy estimates, and then proceed to the discrete setting. We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method. By applying the discrete energy method and imitating the continuous analysis, the discrete estimate that resembles the continuous counterpart is obtained proving stability. We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator. Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented. The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier–Stokes equations.

A stable FDTD subgridding scheme with SBP-SAT for transient TM analysis

We proposed a provably stable FDTD subgridding method for accurate and efficient transient electromagnetic analysis. In the proposed method, several field components are properly added to the boundaries of Yee's grid to make sure that the discrete operators meet the 2nd-order and 4th-order summation-by-parts (SBP) properties. Then, by incorporating the simultaneous approximation terms (SATs) into the finite-difference time-domain (FDTD) method, the proposed FDTD subgridding method mimics the energy estimate of the continuous Maxwell's equations at the semi-discrete level to guarantee its stability. Furthermore, to couple multiple mesh blocks with different mesh sizes, the interpolation matrices are also derived. The proposed FDTD subgridding method is accurate, efficient, easy to implement and integrate into the existing FDTD codes with only simple modifications. At last, four numerical examples with fine structures are carried out to validate the effectiveness of the proposed method.

Simulation of flexural-gravity wave propagation for elastic plates in shallow water using an energy-stable finite difference method with weakly enforced boundary and interface conditions

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.

Non-conforming Interface Conditions for the Second-Order Wave Equation

Imposition methods of interface conditions for the second-order wave equation with non-conforming grids is considered. The spatial discretization is based on high order finite differences with summation-by-parts properties. Previously presented solution methods for this problem, based on the simultaneous approximation term (SAT) method, have shown to introduce significant stiffness. This can lead to highly inefficient schemes. Here, two new methods of imposing the interface conditions to avoid the stiffness problems are presented: (1) a projection method and (2) a hybrid between the projection method and the SAT method. Numerical experiments are performed using traditional and order-preserving interpolation operators. Both of the novel methods retain the accuracy and convergence behavior of the previously developed SAT method but are significantly less stiff.

Modeling and inversion in acoustic-elastic coupled media using energy-stable summation-by-parts operators

Seismic data acquired at the seafloor are valuable in characterizing the subsurface and monitoring producing hydrocarbon fields. To fully use such data, a stable, accurate, and efficient numerical scheme is needed that accounts for acoustic and elastic wave propagation, their interaction at the seafloor interface, and for sources and receivers placed on either side of that interface. Existing methods either make incorrect assumptions or have high implementation and computational costs. We have developed a high-order finite-difference summation-by-parts framework for the acoustic-elastic wave equations in second-order form (in terms of displacements in the solid and an extension of velocity potential in the fluid). Our modified discretization of the elastic operator overcomes the dispersion errors known to plague displacement-based schemes in the high VP/ VS limit. We weakly impose boundary and interface conditions using simultaneous approximation terms, leading to an energy-stable numerical scheme that rigorously handles point injection and extraction. The fully discrete system is self-adjoint after a time reversal and a sign flip and is furthermore a high-order accurate discretization of the continuous problem. The self-adjointness ensures that forward and adjoint wavefields are computed with similar accuracy and simplifies gradient computation for inversion purposes. We find that our numerical scheme achieves accuracy comparable to a more computationally expensive spectral-element method and demonstrate its application to full-waveform inversion using the Marmousi2 model.

An SBP-SAT FDTD Subgridding Method Using Staggered Yee’s Grids Without Modifying Field Components for TM Analysis

A summation-by-parts simultaneous approximation term (SBP-SAT) finite-difference time-domain (FDTD) subgridding method is proposed to model geometrically fine structures in this article. Compared with other works, the proposed SBP-SAT FDTD method uses the staggered Yee’s grid without adding or modifying any field components through field extrapolation on the boundaries to make the discrete operators satisfy the SBP property. The accuracy of extrapolation keeps consistency with that of the second-order finite-difference scheme near the boundaries. In addition, the SATs are used to weakly enforce the tangential boundary conditions between multiple mesh blocks with different mesh sizes. With carefully designed interpolation matrices and selected free parameters of the SATs, no dissipation occurs in the whole computational domain. Therefore, its long-time stability is theoretically guaranteed. Four numerical examples are carried out to validate its effectiveness. Results show that the proposed SBP-SAT FDTD subgridding method is stable, accurate, efficient, and easy to implement based on the existing FDTD codes with only a few modifications.

A Non-Stiff Summation-By-Parts Finite Difference Method for the Scalar Wave Equation in Second Order Form: Characteristic Boundary Conditions and Nonlinear Interfaces

Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard method can become arbitrarily stiff when characteristic boundary conditions and nonlinear interface conditions are used. Here we propose a new technique that avoids this stiffness by using characteristic variables to “upwind” the boundary and interface treatment. This is done through the introduction of an additional block boundary displacement variable. Using a unified energy, which expresses both the standard as well as characteristic boundary and interface treatment, we show that the resulting scheme has semidiscrete energy stability for the scalar anisotropic wave equation. The theoretical stability results are confirmed with numerical experiments that also demonstrate the accuracy and robustness of the proposed scheme. The numerical results also show that the characteristic scheme has a time step restriction based on standard wave propagation considerations and not the boundary closure.

Entropy stability for the compressible Navier-Stokes equations with strong imposition of the no-slip boundary condition

We consider the compressible Navier-Stokes equations subject to no-slip adiabatic wall boundary conditions. The main goal is to investigate stability properties of schemes imposing the no-slip condition strongly (injection) and the temperature condition weakly by a simultaneous approximation term. To this end, we propose a low-order summation-by-parts scheme. By verifying the complete linearisation procedure, we prove linear stability for the scheme. In addition, and assuming that the interior scheme is entropy stable, we also prove entropy stability for the full scheme including the boundary treatment. Furthermore, we propose a linearly stable 3rd-order scheme with the same imposition of the wall conditions. However, the 3rd-order scheme is not provably non-linearly stable. A number of simulations show that the boundary procedure is robust for both schemes.

A comparative study of two different shallow water formulations using stable summation by parts schemes

This study provides numerical solutions to the two-dimensional linearized shallow water equations (SWE) using a high-order finite difference scheme in Summation By Parts (SBP) form. In addition to the SBP operators for the discretizations, penalty terms, Simultaneous Approximation Terms (SAT) are applied to impose well-posed open boundary conditions. The conventional SWE with height and velocities as the prognostic variables, and a new type of the vorticity–divergence SWE with wave height gradients, vorticity and divergence as the prognostic variables were investigated. It was shown that the solution in all numerical tests enter and exit the domain without instabilities. The convergence rates were correct for all orders of the SBP operators in both the entrance and exit tests. Interestingly, the error norm of the wave height were orders of magnitude lower in the vorticity–divergence solutions compared to the conventional SWE solutions.

Positivity-preserving entropy stable schemes for the 1-D compressible Navier-Stokes equations: High-order flux limiting

This is the second of two companion papers, in which we develop and analyze a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 1-D compressible Navier-Stokes equations. The key distinctive property of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method developed in the companion paper [1]. This first-order scheme is discretized on the same Legendre-Gauss-Lobatto points used for the high-order counterpart. As a result, no interpolation or restriction is required between high- and low-order elements which are coupled by using the simultaneous approximation term (SAT) penalty method. To provide positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves the design–order of accuracy of the corresponding high-order baseline spectral collocation method for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorous L2-stability proof for the symmetric form of the discretized Navier-Stokes equations. The proposed methodology is very general and can be extended to multiple dimensions and other summation-by-parts SAT-type schemes.

An Energy-Based Summation-by-Parts Finite Difference Method For the Wave Equation in Second Order Form

We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the simultaneous-approximation-term (SAT) method, we derive energy estimates for the semi-discretization. In addition, error estimates are derived by the normal mode analysis. The proposed method is termed as energy-based because of its similarity with the energy-based discontinuous Galerkin method. When imposing the Dirichlet boundary condition and material interface conditions, the traditional SBP-SAT discretization uses a penalty term with a mesh-dependent parameter, which is not needed in our method. Furthermore, numerical dissipation can be added to the discretization through the boundary and interface conditions. We present numerical experiments that verify convergence and robustness of the proposed method.

An entropy–stable p–adaptive nodal discontinuous Galerkin for the coupled Navier–Stokes/Cahn–Hilliard system

We develop a novel entropy–stable discontinuous Galerkin approximation of the incompressible Navier–Stokes/Cahn–Hilliard system for p–non–conforming elements. This work constitutes an evolution of the work presented by Manzanero et al. ((2020) [10]), as it extends the discrete analysis into supporting p–adaptation (p–refinement/coarsening). The scheme is based on the summation–by–parts simultaneous–approximation term property along with Gauss–Lobatto points and suitable numerical fluxes. The p–non–conforming elements are connected through the classic mortar method, the use of central fluxes for the inviscid terms, and the BR1 scheme with additional dissipation for the viscous fluxes. The scheme is proven to retain its properties of the original conforming scheme when transitioning to p–non–conforming elements and to mimic the continuous entropy analysis of the model. We focus on dynamic polynomial adaptation as the applications of interest are unsteady multiphase flows. In this work, we introduce a heuristic adaptation criterion that depends on the location of the interface between the different phases and utilises the convection velocity to predict the movement of the interface. The scheme is verified to be total phase conserving, entropy–stable and freestream preserving for curvilinear p–non–conforming meshes. We also present the results for a rising bubble simulation and we show that for the same accuracy we get a ×2 to ×6 reduction in the degrees of freedom and a 41% reduction in the computational time. We compare our results for the three–dimensional dam break test case against experimental and numerical data and we show that a ×4.3 to ×9.5 reduction of the degrees of freedom and a 51% reduction in the computational time can be achieved compared to the p–uniform solution.

An entropy–stable discontinuous Galerkin approximation of the Spalart–Allmaras turbulence model for the compressible Reynolds Averaged Navier–Stokes equations

We present an entropy–stable formulation for the compressible Reynolds Averaged Navier–Stokes (RANS) Discontinuous Galerkin (DG) equations and the Spalart–Allmaras one–equation closure. The model is designed to satisfy an entropy law, which includes free– and no–slip wall boundary conditions. We then construct a high–order DG approximation of the model that satisfies the summation–by–parts simultaneous–approximation–term (SBP-SAT) property. With the help of a discrete stability analysis, we construct two approximations: a kinetic energy preserving scheme based on Pirozzoli's two–point flux and a thermodynamic entropy conserving one based on Chandrashekar's split–form. The schemes are applicable to, and the stability proofs hold for, three–dimensional unstructured meshes with curvilinear hexahedral elements. We test the convergence of the schemes on a manufactured solution for increasing polynomial orders and mesh refinement levels, to then assess their numerical stability by propagating a flow from a random initial condition, and finally solve the flow around a two–dimensional flat plate and a NACA 0012 airfoil, comparing numerical results with those available in the literature. The proposed schemes are entropy–stable, and provide accurate solutions for the selected test cases.

Stability and Functional Superconvergence of Narrow-Stencil Second-Derivative Generalized Summation-By-Parts Discretizations

We analyze the stability and functional superconvergence of discretizations of diffusion problems with the narrow-stencil second-derivative generalized summation-by-parts (SBP) operators coupled with simultaneous approximation terms (SATs). Provided that the primal and adjoint solutions are sufficiently smooth and the SBP-SAT discretization is primal and adjoint consistent, we show that linear functionals associated with the steady diffusion problem superconverge at a rate of $ 2p $ when a degree $ p+1 $ narrow-stencil or a degree $ p $ wide-stencil generalized SBP operator is used for the spatial discretization. Sufficient conditions for stability of adjoint consistent discretizations with the narrow-stencil generalized SBP operators are presented. The stability analysis assumes nullspace consistency of the second-derivative operator and the invertibility of the matrix approximating the first derivative at the element boundaries. The theoretical results are verified by numerical experiments with the one-dimensional Poisson problem.

Inverses of SBP-SAT Finite Difference Operators Approximating the First and Second Derivative

The scalar, one-dimensional advection equation and heat equation are considered. These equations are discretized in space, using a finite difference method satisfying summation-by-parts (SBP) properties. To impose the boundary conditions, we use a penalty method called simultaneous approximation term (SAT). Together, this gives rise to two semi-discrete schemes where the discretization matrices approximate the first and the second derivative operators, respectively. The discretization matrices depend on free parameters from the SAT treatment. We derive the inverses of the discretization matrices, interpreting them as discrete Green’s functions. In this direct way, we also find out precisely which choices of SAT parameters that make the discretization matrices singular. In the second derivative case, it is shown that if the penalty parameters are chosen such that the semi-discrete scheme is dual consistent, the discretization matrix can become singular even when the scheme is energy stable. The inverse formulas hold for SBP-SAT operators of arbitrary order of accuracy. For second and fourth order accurate operators, the inverses are provided explicitly.

Simultaneous approximation terms and functional accuracy for diffusion problems discretized with multidimensional summation-by-parts operators

Several types of simultaneous approximation term (SAT) for diffusion problems discretized with diagonal-norm multidimensional summation-by-parts (SBP) operators are analyzed based on a common framework. Conditions under which the SBP-SAT discretizations are consistent, conservative, adjoint consistent, and energy stable are presented. For SATs leading to primal and adjoint consistent discretizations, the error in output functionals is shown to be of order h2p when a degree p multidimensional SBP operator is used to discretize the spatial derivatives. SAT penalty coefficients corresponding to various discontinuous Galerkin fluxes developed for elliptic partial differential equations are identified. We demonstrate that the original method of Bassi and Rebay, the modified method of Bassi and Rebay, and the symmetric interior penalty method are equivalent when implemented with SBP diagonal-E operators that have diagonal norm matrix, e.g., the Legendre-Gauss-Lobatto SBP operator in one space dimension. Similarly, the local discontinuous Galerkin and the compact discontinuous Galerkin schemes are equivalent for this family of operators. The analysis remains valid on curvilinear grids if a degree ≤p+1 bijective polynomial mapping from the reference to physical elements is used. Numerical experiments with the two-dimensional Poisson problem support the theoretical results.

A Low Dissipative and Stable Cell-Centered Finite Volume Method with the Simultaneous Approximation Term for Compressible Turbulent Flows

A simultaneous-approximation term is a non-reflecting boundary condition that is usually accompanied by summation-by-parts schemes for provable time stability. While a high-order convective flux based on reconstruction is often employed in a finite-volume method for compressible turbulent flow, finite-volume methods with the summation-by-parts property involve either equally weighted averaging or the second-order central flux for convective fluxes. In the present study, a cell-centered finite-volume method for compressible Naiver–Stokes equations was developed by combining a simultaneous-approximation term based on extrapolation and a low-dissipative discretization method without the summation-by-parts property. Direct numerical simulations and a large eddy simulation show that the resultant combination leads to comparable non-reflecting performance to that of the summation-by-parts scheme combined with the simultaneous-approximation term reported in the literature. Furthermore, a characteristic boundary condition was implemented for the present method, and its performance was compared with that of the simultaneous-approximation term for a direct numerical simulation and a large eddy simulation to show that the simultaneous-approximation term better maintained the average target pressure at the compressible flow outlet, which is useful for turbomachinery and aerodynamic applications, while the characteristic boundary condition better preserved the flow field near the outlet.