Sigrun Ortleb

A positivity preserving and well-balanced DG scheme using finite volume subcells in almost dry regions

This paper deals with a subcell finite volume strategy for a discontinuous Galerkin (DG) scheme discretizing the shallow water equations with non-constant bottom topography. Using finite volume subcells within the DG scheme is a recent shock capturing strategy decomposing elements marked by a shock indicator into a specific number of subcells on which a robust FV scheme is applied. This work newly considers the subcell approach in the context of wetting and drying shallow water flows. Here, finite volume subcells are introduced for almost dry cells in order to enable an improved resolution of the wet/dry front. Furthermore, in this work, the theory for positivity preserving and well-balanced DG schemes is extended to the DG scheme using finite volume subcells. The performance of the resulting novel approach both in the context of preserving lake at rest stationary states and for wetting and drying computations is then numerically demonstrated for challenging standard test cases in two space dimensions on unstructured triangular grids.

A Kinetic Energy Preserving DG Scheme Based on Gauss---Legendre Points

In the context of numerical methods for conservation laws, not only the preservation of the primary conserved quantities can be of interest, but also the balance of secondary ones such kinetic energy in case of the Euler equations of gas dynamics. In this work, we construct a kinetic energy preserving discontinuous Galerkin method on Gauss–Legendre nodes based on the framework of summation-by-parts operators. For a Gauss–Legendre point distribution, boundary terms require special attention. In fact, stability problems will be demonstrated for a combination of skew-symmetric and boundary terms that disagrees with exclusively interior nodal sets. We will theoretically investigate the required form of the corresponding boundary correction terms in the skew-symmetric formulation leading to a conservative and consistent scheme. In numerical experiments, we study the order of convergence for smooth solutions, the kinetic energy balance and the behaviour of different variants of the scheme applied to an acoustic pressure wave and a viscous shock tube. Using Gauss–Legendre nodes results in a more accurate approximation in our numerical experiments for viscous compressible flow. Moreover, for two-dimensional decaying homogeneous turbulence, kinetic energy preservation yields a better representation of the energy spectrum.

Short Note: A comparison of the Discontinuous-Galerkin- and Spectral-Difference-Method on triangulations using PKD polynomials

This paper gives a first numerical comparison of the Discontinuous-Galerkin-(DG) and Spectral-Difference-Method (SD) under almost equivalent conditions for smooth test cases.

The DG Scheme on Triangular Grids with Adaptive Modal and Variational Filtering Routines Applied to Shallow Water Flows

The shallow water equations with non-flat bottom topography may describe flows in rivers, lakes or coastal areas. It is well known that this system of balance laws admits discontinuous solutions and numerical schemes have to account for this difficulty. In this contribution, we use the discontinuous Galerkin method to solve these equations. In order to introduce a small but sufficient amount of numerical dissipation to the scheme, we apply a spectral viscosity based damping strategy developed in [10, 11]. This strategy consists of efficient adaptive modal filtering which is directly applied to the coefficients of the numerical solution. In the context of non-flat bottom topography, an extra challenge is posed by steady states with non-zero flux gradients that are exactly balanced by the non-zero source term, hence well-balancedness is required. In addition, non-negativity of the water height has to be preserved. In this contribution, we extend the work of Xing, Zhang and Shu [18] regarding positivity preservation and well-balancedness to triangulations but stay with filtering procedures as our shock capturing strategy.

Patankar-type Runge-Kutta schemes for linear PDEs

textabstractWe study the local discretization error of Patankar-type Runge-Kutta methods applied to semi-discrete PDEs. For a known two-stage Patankar-type scheme the local error in PDE sense for linear advection or diffusion is shown to be of the maximal order

Adopting (s)EPIRK schemes in a domain-based IMEX setting

{\cal O}(\Delta t^3)

On Stability and Conservation Properties of (s)EPIRK Integrators in the Context of Discretized PDEs

for sufficiently smooth and positive exact solutions. However, in a test case mimicking a wetting-drying situation as in the context of shallow-water flows, this scheme yields large errors in the drying region. A more realistic approximation is obtained by a modification of the Patankar approach incorporating an explicit testing stage into the implicit trapezoidal rule.

An application of parameter sensitivity analysis for shallow water flow around a pier

The simulation of viscous, compressible flows around complex geometries or similar applications often inherit the task of solving large, stiff systems of ODEs. Domain-based implicit-explicit (IMEX) type schemes offer the possibility to apply two different schemes to different parts of the computational domain. The goal hereby is to decrease the computational cost by increasing the admissible step sizes with no loss of stability and by reducing the system sizes of the linear solver within the implicit integrator. But which combination of methods reaches the largest gain in efficiency? Coupling of Runge-Kutta methods or different multistep methods has been investigated so far by other authors. Here, we inspect the adoption of the recently introduced exponential integrators called EPIRK and sEPIRK in the IMEX setting, since they are perfectly suited for large, stiff systems of ODEs.

Positivity Preserving Implicit and Partially Implicit Time Integration Methods in the Context of the DG Scheme Applied to Shallow Water Flows

Exponential integrators are becoming increasingly popular for stiff problems of high dimension due to their attractive property of solving the linear part of the system exactly and hence being A-stable. In practice, however, exponential integrators are implemented using approximation techniques to matrix-vector products involving functions of the matrix exponential (the so-called \(\varphi \)-functions) to make them efficient and competitive to other state-of-the-art schemes. We will examine linear stability and provide a Courant–Friedrichs–Lewy (CFL) condition of special classes of exponential integrator schemes called EPIRK and sEPIRK and demonstrate their dependence on the parameters of the embedded approximation technique. Furthermore, a conservation property of the EPIRK schemes is proven.

Adaptive Spectral Filtering and Digital Total Variation Postprocessing for the DG Method on Triangular Grids: Application to the Euler Equations

This work deals with the numerical simulation of shallow water flows in the context of practical applications. The shallow water equations including non-constant bottom topography and bottom friction are discretized in space by the DG scheme while for time integration an implicit unconditionally positivity preserving Runge-Kutta type scheme has been developed. With respect to applications, the focus of this work is on the interpretation of numerical results in the situation of uncertain input data, e.g. experimentally determined data. In this context, a global sensitivity analysis based on an ANOVA decomposition yields information on the impact of given input parameters on the variance of the numerical solution with respect to model parameters as well as initial and boundary conditions. The application of this method is studied for shallow water flow around a pier with uncertain input data of water height, discharge and Manning coefficient in the friction term.