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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. Funding: BAJ acknowledges support from the National Research Foundation of South Africa under Grant Numbers 129119 and 127567. FL and JN were supported by Vetenskapsrådet, Sweden Grant Numbers 2018-05084 and 2021-05484.

In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant-Friedrichs-Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superconvergent linear functionals. Funding Agencies|Linkoping University; Southern California Earthquake Center [10148]; National Science FoundationNational Science Foundation (NSF) [EAR-1600087]; United States Geological SurveyUnited States Geological Survey [G17AC00047]; NSF-ACI Award [1450451]; Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]

We combine existing summation-by-parts discretization methods to obtain a simplified numerical framework for partial differential equations posed on complex multi-block/element domains. The interfaces (conforming or non-conforming) between blocks are treated with inner-product-preserving interpolation operators, and the result is a high-order multi-block operator on summation-by-parts form that encapsulates both the metric terms as well as the interface treatments. This enables for a compact description of the numerical scheme that mimics the essential features of its continuous counterpart. Furthermore, the stability analysis on a multi-block domain is simplified for both for linear and nonlinear equations, since no problem-specific interface conditions need to be derived and implemented. We exemplify the combined operator technique by considering a nonlinearly stable discrete formulation of the incompressible Navier-Stokes equations and perform calculations on an underlying multi-block domain. Funding: Vetenskapsradet, Sweden [2020-03642, 2018-05084, 2021-05484]; Swedish e-Science Research Center (SeRC)

We use the energy method to study the well-posedness of initial-boundary value problems approximated by overset mesh methods in one and two space dimensions for linear constant-coefficient hyperbolic systems. We show that in one space dimension, for both scalar equations and systems of equations, the problem where one domain partially oversets another is well-posed when characteristic coupling conditions are used. If a system cannot be diagonalized, as is usually the case in multiple space dimensions, then the energy method does not give proper bounds in terms of initial and boundary data. For those problems, we propose a novel penalty approach. We show, by using a global energy that accounts for the energy in the overlap region of the domains, that under well-defined conditions on the coupling matrices the penalized overset domain problems are energy bounded, conservative, well-posed and have solutions equivalent to the original single domain problem. Funding: Simons Foundation [426393]; Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC); Klaus-Tschira Stiftung; European Research CouncilEuropean Research Council (ERC)European Commission [71448]

The influence of different boundary conditions on the spectral properties of the incompressible Navier-Stokes equations is investigated. By using the Fourier-Laplace transform technique, we determine the spectra, extract the decay rate in time, and investigate the dispersion relation. In contrast to an infinite domain, where only diffusion affects the convergence, we show that also the propagation speed influence the rate of convergence to steady state for a bounded domain. Once the continuous equations are analyzed, we discretize using high-order finite-difference operators on summation-by-parts form and demonstrate that the continuous analysis carries over to the discrete setting. The theoretical results are verified by numerical experiments, where we highlight the necessity of high accuracy for a correct description of time-dependent phenomena. Funding agency: The Swedish e-Science Research Centre (SeRC)