Hybrid Entropy Stable HLL-Type Riemann Solvers for Hyperbolic Conservation Laws

It is known that HLL-type schemes are more dissipative than schemes based on characteristic decompositions. However, HLL-type methods offer greater flexibility to large systems of hyperbolic conservation laws because the eigenstructure of the flux Jacobian is not needed. We demonstrate in the present work that several HLL-type Riemann solvers are provably entropy stable. Further, we provide convex combinations of standard dissipation terms to create hybrid HLL-type methods that have less dissipation while retaining entropy stability. The decrease in dissipation is demonstrated for the ideal MHD equations with a numerical example.Comment: 6 page

Third-order Limiting for Hyperbolic Conservation Laws applied to Adaptive Mesh Refinement and Non-Uniform 2D Grids

In this paper we extend the recently developed third-order limiter function $H_{3\text{L}}^{(c)}$ [J. Sci. Comput., (2016), 68(2), pp.~624--652] to make it applicable for more elaborate test cases in the context of finite volume schemes. This work covers the generalization to non-uniform grids in one and two space dimensions, as well as two-dimensional Cartesian grids with adaptive mesh refinement (AMR). The extension to 2D is obtained by the common approach of dimensional splitting. In order to apply this technique without loss of third-order accuracy, the order-fix developed by Buchm\"uller and Helzel [J. Sci. Comput., (2014), 61(2), pp.~343--368] is incorporated into the scheme. Several numerical examples on different grid configurations show that the limiter function $H_{3\text{L}}^{(c)}$ maintains the optimal third-order accuracy on smooth profiles and avoids oscillations in case of discontinuous solutions

On Third-Order Limiter Functions for Finite Volume Methods

In this article, we propose a finite volume limiter function for a reconstruction on the three-point stencil. Compared to classical limiter functions in the MUSCL framework, which yield $2^{\text{nd}}$-order accuracy, the new limiter is $3^\text{rd}$-order accurate for smooth solutions. In an earlier work, such a $3^\text{rd}$-order limiter function was proposed and showed successful results [2]. However, it came with unspecified parameters. We close this gap by giving information on these parameters.Comment: 8 pages, conference proceeding

Hybrid Riemann Solvers for Large Systems of Conservation Laws

In this paper we present a new family of approximate Riemann solvers for the numerical approximation of solutions of hyperbolic conservation laws. They are approximate, also referred to as incomplete, in the sense that the solvers avoid computing the characteristic decomposition of the flux Jacobian. Instead, they require only an estimate of the globally fastest wave speeds in both directions. Thus, this family of solvers is particularly efficient for large systems of conservation laws, i.e. with many different propagation speeds, and when no explicit expression for the eigensystem is available. Even though only fastest wave speeds are needed as input values, the new family of Riemann solvers reproduces all waves with less dissipation than HLL, which has the same prerequisites, requiring only one additional flux evaluation.Comment: 9 page

Relations between WENO3 and Third-order Limiting in Finite Volume Methods

Weighted essentially non-oscillatory (WENO) and finite volume (FV) methods employ different philosophies in their way to perform limiting. We show that a generalized view on limiter functions, which considers a two-dimensional, rather than a one-dimensional dependence on the slopes in neighboring cells, allows to write WENO3 and $3^\text{rd}$-order FV schemes in the same fashion. Within this framework, it becomes apparent that the classical approach of FV limiters to only consider ratios of the slopes in neighboring cells, is overly restrictive. The hope of this new perspective is to establish new connections between WENO3 and FV limiter functions, which may give rise to improvements for the limiting behavior in both approaches.Comment: 22 page

On building blocks of finite volume methods : Limiter functions and Riemann solvers

In this thesis we are interested in numerically solving conservation laws with high-order finite volume methods. Hyperbolic systems of partial differential equations are especially challenging since even smooth initial flows may develop discontinuities in finite time. Naively discretizing such flows with high-order schemes may lead to undesired oscillations at discontinuities. First-order methods do not encounter this problem since shocks are smeared out. Nevertheless, high-order schemes are in demand because they have the advantage of reaching a fixed error bound on coarser grids than low-order methods. This reduces the overall computational time and thus the total cost. Combining the advantages of several methods, limiter functions change from high- to low-order whenever necessary. This avoids oscillations at discontinuities while maintaining high-order accuracy at smooth parts of the solution. Thus, the resulting schemes are applicable to physically relevant problems which often contain smooth parts as well as large gradients, discontinuities, or shocks. The aim of this work is the development of third-order finite volume methods by improving building blocks of the method. This means, we identify main routines in the finite volume framework and present new concepts for improving their performance. We focus on two building blocks. First, the high-order reconstruction of interface values using limiter functions. Second, the numerical flux function, also referred to as Riemann solver. The former is crucial for the order of accuracy of the solution while the latter determines the amount of dissipation added to the scheme. We develop a new third-order accurate reconstruction function for the spatial approximation of hyperbolic conservation laws. This reconstruction switches between first- and third-order, resulting in a scheme which is high-order accurate in smooth parts of the solution, does not create oscillations at discontinuities, and avoids extrema clipping as encountered by total variation diminishing (TVD) methods. The novel limiter only needs information from the cell of interest and its nearest neighbors, thus keeping the stencil as compact as possible for obtaining third order accuracy. Furthermore, the reconstruction remains in the traditional second-order framework, easing the implementation of the limiter in existing codes. Finally, a decision criterion without artificial parameters is incorporated in the limiter. This decision criterion distinguishes shocks and large gradients from extrema, thus ensuring accurate shock capturing. The obtained reconstructions at each side of the cell boundaries are then inserted into the numerical flux function which solves local Riemann problems. Many numerical flux functions, also referred to as Riemann solvers, have been developed over the last decades. However, most classical solvers add too much dissipation to the scheme such that discontinuities are smeared out. On the other side of the spectrum, Riemann solvers that do not add too much dissipation need information on the eigen structure which is costly to compute for large systems. There is the need for new Riemann solvers that avoid solving for the eigen system and still reproduce all waves of the system with less dissipation than classical methods such as Rusanov and Harten-Lax-van Leer (HLL). We present a hybrid family of Riemann solvers, requiring only an estimate of the globally fastest wave speeds in both directions. Thus, the new solvers are particularly efficient for large systems of conservation laws when no explicit expression for the eigen system is available or expensive to compute. For the validation of the developed schemes we conduct a series of numerical experiments. First, we demonstrate that the novel high-order limiter function obtains the desired third-order accuracy for smooth solutions. Test cases includes mooth and discontinuous linear transport, Euler equations, and ideal magneto hydrodynamics (MHD). Problems are presented in one and two space dimensions, on uniform as well as non-uniform grids and with adaptive mesh refinement. In a second step, the hybrid family of Riemann solvers is tested in a first-order framework. Here, we show that the newly developed solvers induce less dissipation than schemes with comparable input data. This leads to sharper gradients and less smearing at discontinuities. Numerical examples contain linear transport, Euler equations, ideal MHD, as well as the regularized 13-moment equations (R13).Finally, both parts of this work are combined to obtain third-order accurate results. We reconstruct using the novel third-order limiter and insert there constructed interface values into the hybrid family of Riemann solvers. For all numerical examples, we also implement comparable methods to ascertain the quality of our schemes. The solutions obtained with the newly developed methods indeed indicate better or equally-good results and an excellent performance

On third-order limiter functions for finite volume methods

In this article, we propose a finite volume limiter function for a reconstruction on the three-point stencil. Compared to classical limiter functions in the MUSCL framework, which yield 2$^{rd}$-order accuracy, the new limiter is 3$^{rd}$-order accurate for smooth solution. In an earlier work, such a 3$^{rd}$-order limiter function was proposed and showed successful results [2]. However, it came with unspecified parameters. We close this gap by giving information on these parameters