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Finite-difference based approaches are common for approximating the Caputo fractional derivative. Often, these methods lead to a reduction of the convergence rate that depends on the fractional order. In this note, we approximate the expressions in the fractional derivative components using a separate quadrature rule for the integral and a separate discretization of the derivative in the integrand. By this approach, the error terms from the Newton–Cotes quadrature and the differentiation are isolated and it is possible to conclude that the order dependent error is inevitable when the end points of the interval are included in the quadrature. Furthermore, we show experimentally that the theoretical findings carries over to quadrature rules without the end points included. Finally we show how to increase accuracy for smooth functions, and compensate for the order dependent loss. [ABSTRACT FROM AUTHOR]

Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions. [ABSTRACT FROM AUTHOR]

We use the framework of upwind summation-by-parts (SBP) operators developed by Mattsson (2017, doi:10.1016/j.jcp.2017.01.042) and study different flux vector splittings in this context. To do so, we introduce discontinuous-Galerkin-like interface terms for multi-block upwind SBP methods applied to nonlinear conservation laws. We investigate the behavior of the upwind SBP methods for flux vector splittings of varying complexity on Cartesian as well as unstructured curvilinear multi-block meshes. Moreover, we analyze the local linear/energy stability of these methods following Gassner, Svärd, and Hindenlang (2022, doi:10.1007/s10915-021-01720-8). Finally, we investigate the robustness of upwind SBP methods for challenging examples of shock-free flows of the compressible Euler equations such as a Kelvin-Helmholtz instability and the inviscid Taylor-Green vortex. • Different flux vector splitting schemes for upwind summation-by-parts methods. • Discontinuous-Galerkin-like interface terms for upwind summation-by-parts methods. • Investigate free-stream preservation on curvilinear unstructured quad meshes. • Study local linear/energy stability properties. • Investigate impact on robustness for under-resolved simulations. [ABSTRACT FROM AUTHOR]

Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future. • We introduce second-derivative summation-by-parts operators for general (non-polynomial) function spaces. • We prove their existence and mimetic properties. • We present a straightforward construction procedure for general multi-dimensional domains/elements/blocks. [ABSTRACT FROM AUTHOR]

Diagonal norm finite difference based time integration methods in summation-byparts form are investigated. The second, fourth, and sixth order accurate discretizations are proven to have eigenvalues with strictly positive real parts. This leads to provably invertible fully discrete approximations of initial boundary value problems. Our findings also allow us to conclude that the Runge--Kutta methods based on second, fourth, and sixth order summation-by-parts finite difference time discretizations automatically satisfy previously unreported stability properties. The procedure outlined in this article can be extended to even higher order summation-by-parts approximations with repeating stencil. [ABSTRACT FROM AUTHOR]

We simulate earthquake cycles with rate-and-state fault friction and off-fault power-law viscoelasticity for the classic 2D antiplane shear problem of a vertical, strike-slip plate boundary fault. We investigate the interaction between fault slip and bulk viscous flow with experimentally-based flow laws for quartz-diorite and olivine for the crust and mantle, respectively. Simulations using three linear geotherms ( dT / dz = 20, 25, and 30 K/km) produce different deformation styles at depth, ranging from significant interseismic fault creep to purely bulk viscous flow. However, they have almost identical earthquake recurrence interval, nucleation depth, and down-dip coseismic slip limit. Despite these similarities, variations in the predicted surface deformation might permit discrimination of the deformation mechanism using geodetic observations. Additionally, in the 25 and 30 K/km simulations, the crust drags the mantle; the 20 K/km simulation also predicts this, except within 10 km of the fault where the reverse occurs. However, basal tractions play a minor role in the overall force balance of the lithosphere, at least for the flow laws used in our study. Therefore, the depth-integrated stress on the fault is balanced primarily by shear stress on vertical, fault-parallel planes. Because strain rates are higher directly below the fault than far from it, stresses are also higher. Thus, the upper crust far from the fault bears a substantial part of the tectonic load, resulting in unrealistically high stresses. In the real Earth, this might lead to distributed plastic deformation or formation of subparallel faults. Alternatively, fault pore pressures in excess of hydrostatic and/or weakening mechanisms such as grain size reduction and thermo-mechanical coupling could lower the strength of the ductile fault root in the lower crust and, concomitantly, off-fault upper crustal stresses. [ABSTRACT FROM AUTHOR]

Fully-implicit discrete formulations in summation-by-parts form for initial-boundary value problems must be invertible in order to provide well functioning procedures. We prove that, under mild assumptions, pseudo-spectral collocation methods for the time derivative lead to invertible discrete systems when energy-stable spatial discretizations are used. [ABSTRACT FROM AUTHOR]

In this work, we propose an extension of telescopic derivative operators for the DGSEM with Gauss nodes, and we prove that this formulation is equivalent to its usual matrix counterpart. Among other possible applications, this allows extending the stabilization methods already developed for Gauss-Lobatto nodes to Gauss nodes, also ensuring properties such as entropy stability while retaining their improved accuracy. [ABSTRACT FROM AUTHOR]

The advection equation is studied in a completely general two domain setting with different wave-speeds and a time-independent jump-condition at the interface separating the domains. Well-posedness and conservation criteria are derived for the initial-boundary-value problem. The equations are semi-discretized using a finite difference method on Summation-By-Part (SBP) form. The relation between the stability and conservation properties of the approximation are studied when the boundary and interface conditions are weakly imposed by the Simultaneous-Approximation-Term (SAT) procedure. Numerical simulations corroborate the theoretical findings. [ABSTRACT FROM AUTHOR]

We analyze the extension of summation-by-parts operators and weak boundary conditions for solving initial boundary value problems involving second derivatives in time. A wide formulation is obtained by first rewriting the problem on first order form. This formulation leads to optimally sharp fully discrete energy estimates that are unconditionally stable and high order accurate. Furthermore, it provides a natural way to impose mixed boundary conditions of Robin type, including time and space derivatives. We apply the new formulation to the wave equation and derive optimal fully discrete energy estimates for general Robin boundary conditions, including nonreflecting ones. The scheme utilizes wide stencil operators in time, whereas the spatial operators can have both wide and compact stencils. Numerical calculations verify the stability and accuracy of the method. We also include a detailed discussion on the added complications when using compact operators in time and give an example showing that an energy estimate cannot be obtained using a standard second order accurate compact stencil. [ABSTRACT FROM AUTHOR]

The accurate and fast prediction of potential propagation in neuronal networks is of prime importance in neurosciences. This work develops a novel structure-preserving model reduction technique to address this problem based on Galerkin projection and nonnegative operator approximation. It is first shown that the corresponding reduced-order model is guaranteed to be energy stable, thanks to both the structure-preserving approach that constructs a distinct reducedorder basis for each cable in the network and the preservation of nonnegativity. Furthermore, a posteriori error estimates are provided, showing that the model reduction error can be bounded and controlled. Finally, the application to the model reduction of a large-scale neuronal network underlines the capability of the proposed approach to accurately predict the potential propagation in such networks while leading to important speedups. [ABSTRACT FROM AUTHOR]

We present a well-posed stochastic Galerkin formulation of the incompressible Navier–Stokes equations with uncertainty in model parameters or the initial and boundary conditions. The stochastic Galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions, resulting in an extended system of equations. A relatively low-order generalized polynomial chaos expansion is sufficient to capture the stochastic solution for the problem considered. We derive boundary conditions for the continuous form of the stochastic Galerkin formulation of the velocity and pressure equations. The resulting problem formulation leads to an energy estimate for the divergence. With suitable boundary data on the pressure and velocity, the energy estimate implies zero divergence of the velocity field. Based on the analysis of the continuous equations, we present a semi-discretized system where the spatial derivatives are approximated using finite difference operators with a summation-by-parts property. With a suitable choice of dissipative boundary conditions imposed weakly through penalty terms, the semi-discrete scheme is shown to be stable. Numerical experiments in the laminar flow regime corroborate the theoretical results and we obtain high-order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined. [ABSTRACT FROM AUTHOR]

A new high order Arakawa-like method for the incompressible vorticity equation in two-dimensions has been developed. Mimetic properties such as skew-symmetry, energy and enstrophy conservations for the semi-discretization are proved for periodic problems using arbitrary high order summation-by-parts operators. Numerical simulations corroborate the theoretical findings. [ABSTRACT FROM AUTHOR]

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations. [ABSTRACT FROM AUTHOR]

Abstract: A detailed account of the stability and accuracy properties of the SBP-SAT technique for numerical time integration is presented. We show how the technique can be used to formulate both global and multi-stage methods with high order of accuracy for both stiff and non-stiff problems. Linear and non-linear stability results, including A-stability, L-stability and B-stability are proven using the energy method for general initial value problems. Numerical experiments corroborate the theoretical properties. [Copyright &y& Elsevier]

Abstract: Consider the discretization of a partial differential equation (PDE) and an integral functional that depends on the PDE solution. The discretization is dual consistent if it leads to a discrete dual problem that is a consistent approximation of the corresponding continuous dual problem. Consequently, a dual-consistent discretization is a synthesis of the so-called discrete-adjoint and continuous-adjoint approaches. We highlight the impact of dual consistency on summation-by-parts (SBP) finite-difference discretizations of steady-state PDEs; specifically, superconvergent functionals and accurate functional error estimates. In the case of functional superconvergence, the discrete-adjoint variables do not need to be computed, since dual consistency on its own is sufficient. Numerical examples demonstrate that dual-consistent schemes significantly outperform dual-inconsistent schemes in terms of functional accuracy and error-estimate effectiveness. The dual-consistent and dual-inconsistent discretizations have similar computational costs, so dual consistency leads to improved efficiency. To illustrate the dual consistency analysis of SBP schemes, we thoroughly examine a discretization of the Euler equations of gas dynamics, including the treatment of the boundary conditions, numerical dissipation, interface penalties, and quadrature by SBP norms. [Copyright &y& Elsevier]

Abstract: We develop a new high order accurate time-integration technique for initial value problems. We focus on problems that originate from a space approximation using high order finite difference methods on summation-by-parts form with weak boundary conditions, and extend that technique to the time-domain. The new time-integration method is global, high order accurate, unconditionally stable and together with the approximation in space, it generates optimally sharp fully discrete energy estimates. In particular, it is shown how stable fully discrete high order accurate approximations of the Maxwells’ equations, the elastic wave equations and the linearized Euler and Navier–Stokes equations can obtained. Even though we focus on finite difference approximations, we stress that the methodology is completely general and suitable for all semi-discrete energy-stable approximations. Numerical experiments show that the new technique is very accurate and has limited order reduction for stiff problems. [Copyright &y& Elsevier]

Abstract: The stochastic Galerkin and collocation methods are used to solve an advection–diffusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection–diffusion equation onto the stochastic basis functions. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system. It is essential that the eigenvalues of the resulting viscosity matrix of the stochastic Galerkin system are positive and we investigate conditions for this to hold. When the viscosity matrix is diagonalizable, stochastic Galerkin and stochastic collocation are similar in terms of computational cost, and for some cases the accuracy is higher for stochastic Galerkin provided that monotonicity requirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steady-state. [Copyright &y& Elsevier]

Abstract: Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. The SBP operator definition includes a weight matrix that is used formally for discrete integration; however, the accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. For diagonal weight matrices, the accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; in particular, the diagonal norm accurately approximates the norm for functions, and multi-dimensional SBP discretizations accurately approximate the divergence theorem. [Copyright &y& Elsevier]

Abstract: The paper develops a posteriori error estimates of integral output functionals for summation-by-parts finite-difference methods. The error estimates are based on the adjoint-weighted residual method and take advantage of a variational interpretation of summation-by-parts discretizations. The estimates are computed on a fixed grid and do not require an embedded grid or explicit interpolation operators. For smooth boundary-value problems containing first and second derivatives the error estimates converge to the exact error as the mesh is refined. The theory is verified using linear boundary-value problems and the Euler equations. [Copyright &y& Elsevier]

Diagonal-norm summation-by-parts (SBP) operators call be used to construct time-stable high-order accurate finite-difference schemes. However, to achieve both stability and accuracy, these operators must use s-order accurate boundary closures when the interior scheme is 2s-order accurate. The boundary closure limits the solution to (s + 1)-order global accuracy. Despite this bound on solution accuracy, we show that functional estimates can be constructed that are 2s-order accurate. This superconvergence requires dual-consistency, which depends on the SBP operators, the boundary condition implementation, and the discretized functional. The theory is developed for scalar hyperbolic and elliptic partial differential equations in one dimension. In higher dimensions, we show that superconvergent functional estimates remain viable in the presence of curvilinear multiblock grids with interfaces. The generality of the theoretical results is demonstrated using a two-dimensional Poisson problem and a nonlinear hyperbolic system--the Euler equations of fluid mechanics. [ABSTRACT FROM AUTHOR]

We introduce a hybrid method to couple continuous Galerkin finite element methods and high-order finite difference methods in a nonconforming multiblock fashion. The aim is to optimize computational efficiency when complex geometries are present. The proposed coupling technique requires minimal changes in the existing schemes while maintaining strict stability, accuracy, and energy conservation. Results are demonstrated on linear and nonlinear scalar conservation laws in two spatial dimensions. [ABSTRACT FROM AUTHOR]

New entropy stable spectral collocation schemes of arbitrary order of accuracy are developed for the unsteady 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. To take into account the grid motion and deformation, we use an arbitrary Lagrangian-Eulerian formulation. As a result, moving and deforming hexahedral grid elements are individually mapped onto a cube in the fixed reference system of coordinates. The proposed scheme is constructed by using the skew-symmetric form of the Navier-Stokes equations, which are discretized by using summation-by-parts spectral collocation operators that preserve the conservation properties of the original governing equations. Furthermore, the metric coefficients are approximated such that the geometric conservation laws are satisfied exactly on both static and dynamic grids. To make the scheme entropy stable, a new entropy conservative flux is derived for the 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. The new flux preserves the design order of accuracy of the original spectral collocation scheme and guarantees entropy conservation on moving and deforming grids. We present numerical results demonstrating design order of accuracy and freestream preservation properties of the new schemes for both the Euler and Navier-Stokes equations on dynamic unstructured grids. • New entropy stable SC schemes for the 3D Navier-Stokes equations on dynamic grids. • A new entropy conservative flux for the 3D Navier-Stokes equations on dynamic grids. • A new dual-Jacobian method that satisfies the geometric conservation laws exactly. [ABSTRACT FROM AUTHOR]

We present a novel technique for the imposition of non-linear entropy conservative and entropy stable solid wall boundary conditions for the compressible Navier–Stokes equations in the presence of an adiabatic wall, or a wall with a prescribed heat entropy flow. The procedure relies on the formalism and mimetic properties of diagonal-norm, summation-by-parts and simultaneous-approximation-term operators, and is a generalization of previous works on discontinuous interface coupling [1] and solid wall boundary conditions [2]. Using the method of lines, a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions are coupled with an entropy-conservative or entropy-stable discrete interior operator. The resulting estimate mimics the global entropy estimate obtained at the continuous level. The boundary data at the wall are weakly imposed using a penalty flux approach and a simultaneous-approximation-term technique for both the conservative variables and the gradient of the entropy variables. Discontinuous spectral collocation operators (mass lumped nodal discontinuous Galerkin operators), on high-order unstructured grids, are used for the purpose of demonstrating the robustness and efficacy of the new procedure for weakly enforcing boundary conditions. Numerical simulations confirm the non-linear stability of the proposed technique, with applications to three-dimensional subsonic and supersonic flows. The procedure described is compatible with any diagonal-norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction schemes. [ABSTRACT FROM AUTHOR]