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

  • Komponenten der Finite Volumen Methoden

Schmidtmann, Birte; Torrilhon, Manuel (Thesis advisor); Marquina, Antonio (Thesis advisor)

Aachen (2017, 2018)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017


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.


  • Department of Mathematics [110000]
  • Chair of Applied and Computational Mathematics [115010]