On conservation and stability properties for summation-by-parts schemes

Abstract

High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework. However, if generalised SBP operators are used, the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely -interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain. Here, new formulations are proposed, allowing the creation of discretisations using general SBP operators that are both conservative and stable. Thus, several shortcomings that might be attributed to generalised SBP operators are overcome (cf.There are classical finite difference operators that can be interpreted in an analytical setting similar to the one described above [19]. However, in general -to the authors knowledge -it is not known whether finite difference SBP operators correspond to an analytical basis. Nevertheless, the basic requirement of the SBP property (4) can be enhanced by accuracy conditions in order to get a useful definition of (numerical) SBP operators, cf. [11,59].Definition 1. Using a numerical representation u = (u 0 , . . . , u p ) T of a function u in a nodal basis, the operators D , R , M , and B described above form a qth order SBP discretisation, if 1. the derivative matrix D is exact for polynomials of degree q, 2. the mass / norm matrix M is symmetric and positive definite, 3. the SBP property (4) is fulfilled, and

Energy stable boundary conditions for the nonlinear incompressible Navier–Stokes equations

“…The form of the projection matrices E 0 and E N,M implies that the first and last grid points are located precisely at the boundaries. This choice avoids certain stability problems [41]. Note also that nowhere in the description above is the accuracy of the discrete derivative operators specified, which allows for arbitrary high order accurate schemes.…”

Combining finite element and finite difference methods for isotropic elastic wave simulations in an energy-conserving manner

“…19 One such extension is elastic wave equations with more general constitutive relations, such as those defined on transversely isotropic media. We note here that the finite difference discretized system (25) and the discrete potential energy E D p in (31) are both written formally for general constitutive relations. Meanwhile, for the finite element discretized system (42), the constitutive relation does not appear in the penalty term p associated with the interface treatment, cf.…”

Accuracy of Stable, High-order Finite Difference Methods for Hyperbolic Systems with Non-smooth Wave Speeds

“…To explore convergence rates of high-order accurate SBP-SAT methods we apply the skew-symmetric discretization from [19] to equation (1), which will allow us to obtain a semi-discrete energy estimate mimicking (4). Note that we apply this method without any special procedure applied near points where wave speed a(x) might be non-smooth.…”

“…We again discretize the domain using N + 1 gridpoints, and apply the skewsymmetric SBP-SAT discretization of [19] to (37-39) given by…”

“…SBP methods together with a weak enforcement of boundary conditions through the simultaneous-approximation-term (SAT) technique provide a framework for provably stable numerical discretizations for problems with su cient smoothness properties and exist for a large class of operators including finite di↵erence, finite volume and finite element methods, see [3,24,21] and references therein. We apply an SBP-SAT discretization to the governing equations using a skew-symmetric splitting, which mixes conservative and nonconservative forms, and for smooth problems has the desirable property that the semi-discrete energy rate mimics that of the continuous problem [15,4,19]. The technique of skew-symmetric splitting is widely used when solving a variety of challenging physical problems that involve variable coe cients, nonlinearities and shocks, including the nonlinear Burgers' equation and the shallow water equations [5,22,6,20].…”