Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

In this paper we present a kinetic relaxation scheme for the Euler equations of gas dynamics in one space dimension. The method is easily applicable to solve any complex system of conservation laws. The numerical scheme is based on a relaxation approximation for conservation laws viewed as a discrete velocity model of the Boltzmann equation of kinetic theory. The discrete kinetic equation is solved by a splitting method consisting of a convection phase and a collision phase. The convection phase involves only the solution of linear transport equations and the collision phase instantaneously relaxes the distribution function to an equilibrium distribution. We prove that the first order accurate method is conservative, preserves the positivity of mass density and pressure and entropy stable. An anti-diffusive Chapman-Enskog distribution is used to derive a second order accurate method. The results of numerical experiments on some benchmark problems confirm the efficiency and robustness of the proposed scheme.

We consider Low Mach number gas dynamic flows with significant energy transport. In addition we focus on applications where a one-dimensional description is appropriate. For such flows we discuss various asymptotic regimes in the vanishing Mach number limit. In the first example we study the flow in a chimney. In the second example we investigate the gas dynamics in a so-called Energy Tower, a power plant designed for producing electrical energy using the latent heat of water.

In this work, the accuracy of high order discontinuous Galerkin discretizations for underresolved problems is investigated. Whereas the superior behavior of high order methods for the well resolved case is undisputed, in case of underresolution, the answer is not as clear. The controversy originates from the fact that order of convergence is a concept for discretization parameters tending to zero, whereas underresolution is synonym for large discretization parameters. However, this work shows that even in the case of underresolution, high order discontinuous Galerkin approximations yield superior efficiency compared to their lower order variants due to the better dispersion and dissipation behavior. It is furthermore shown that a very high order accurate discretization (theoretically 16th order in this case) yields even better accuracy than state-of-the-art large eddy simulation methods for the same number of degrees of freedom for the considered example. This result is particularly surprising since those large eddy simulation methods are tuned specifically to capture coarsely resolved turbulence, whereas the considered high order method can be applied directly to a wide range of other multi-scale problems without additional parameter tuning.

Historically, the computation of steady flows has been at the forefront of the development of computational fluid dynamics (CFD). This began with the design of rockets and the computation of the bow shock at supersonic speeds and continued with the aerodynamic design of airplanes at transonic cruising speed [14]. Only in the last decade, increasing focus has been put on unsteady flows, which are more difficult to compute. This has several reasons. First of all, computing power has increased dramatically and for 5,000 Euro it is now possible to obtain a machine that is able to compute about a minute of realtime simulation of a nontrivial unsteady three dimensional flow in a day. As a consequence, ever more nonmilitary companies are able to employ numerical simulations as a standard tool for product development, opening up a large number of additional applications. Examples are the computation of tunnel fires [4], flow around wind turbines [29], fluid-structure-interaction like flutter [10], flows inside nuclear reactors [25], wildfires [24], hurricanes and unsteady weather phenomenas [23], gas quenching [20] and many others. More computing capacities will open up further possibilities in the next decade, which suggests that the improvement of numerical methods for unsteady flows should start in earnest now. Finally, the existing methods for the computation of steady states, while certainly not at the end of their development, have matured, making the consideration of unsteady flows interesting for a larger group of scientists. In this article, we will focus on the computation of laminar viscous flows, as modelled by the Navier-Stokes equations.

In this chapter we consider the numerical solution of the hyperbolic partial differential equations of mathematical morphology in image processing. First we review our completely discrete flux-corrected transport (DFCT) approach. It uses the viscosity form of a specific upwind scheme in order to quantify viscous artifacts. In a subsequent corrector step that viscosity is compensated by a stabilised inverse diffusion step. We present a thorough analysis of the method including a proof of convergence. After that we introduce a useful framework for processing tensor-valued data. Such data appear in important applications in medical image analysis and engineering. We indicate how to extend the DFCT scheme to that setting and present numerical results proving desirable qualities of our method.

In this study, the method of numerical mixing analysis is presented for three-dimensional ocean models with general vertical coordinates. Numerical mixing of a scalar is defined as the decay of the square of the scalar due to the three-dimensional advection discretisation. It is shown that for any advection scheme the numerical mixing can be calculated as the difference between the advected square of the scalar and the square of the advected tracer, divided by the time step. Special emphasis on directional-split advection schemes is made. It is shown that for those directional-split schemes the numerical analysis method is exact only when the involved advection of the square of the scalar is carried out individually for each split step. As applications, an idealised meso-scale eddy test scenario without any explicit mixing is calculated. It is shown that only for high-order advection schemes for the scalar (salinity in that case) and the momentum, a physically reasonable solution is obtained. Finally, the method is demonstrated for a fully realistic application to the dynamics of the Western Baltic Sea. Here it becomes clear that physical and numerical mixing depend on each others (increased physical mixing leads to decreased numerical mixing) with the dynamically most relevant mixing being the effective mixing, i.e., the sum of the physical and the numerical mixing.

The aim of this work is to understand the possible role in the long-term integration of conservative systems of “G-symplectic” methods. It comes out of a collaboration with Dr Adrian Hill of the University of Bath, United Kingdom, and Dr Yousaf Habib of the National University of Science and Technology, Pakistan. Although symplectic behaviour, or the exact conservation of quadratic invariants, for irreducible methods of this type, is not possible [3], there is a G-generalization, similar to the generalization introduced by Dahlquist [6] in the study of non-linear dissipative methods. Specific issues in this research include the role of time-reversal symmetry in conservative integration, the exacerbation of parasitic effects and the construction and implementation of specific methods of increasingly high orders.

The subject of the paper is the numerical simulation of inviscid and viscous compressible flow in time dependent domains. The motion of the boundary of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the Euler and Navier-Stokes equations describing compressible flow. They are discretized in space by the discontinuous Galerkin (DG) finite element method using piecewise polynomial discontinuous approximations. For the time discretization the BDF method or DG in time is used. Moreover, we use a special treatment of boundary conditions and shock capturing, allowing the solution of flow with a wide range of Mach numbers. As a result we get an efficient and robust numerical process. We show that the method allows to solve numerically the flow with a wide range of Mach numbers and it is applicable to the solution of practically relevant problems of flow induced airfoil vibrations.

In this essay the rigorous application of the method of vertical lines, i.e. performing the successive steps of spatial and temporal discretization is investigated for dynamical and quasi-statical systems. A particular focus lies in the field of solid mechanics where constitutive models of evolutionary-type are of basic interest. Various coupled systems, i.e. thermo-mechanical, electro-thermal or electro-thermo-mechanical coupled problems are investigated in view of the structure of their resulting equations, commonly, leading to systems of ordinary differential equations or systems of differential-algebraic equations after the spatial discretization step. For the case of a thermo-mechanical and an electro-thermal problem stiffly accurate diagonally-implicit Runge-Kutta methods are applied.

Multirate schemes for conservation laws or convection-dominated problems seem to come in two flavors: schemes that are locally inconsistent, and schemes that lack mass-conservation. In this paper these two defects are discussed for one-dimensional conservation laws. Particular attention will be given to monotonicity properties of the multirate schemes, such as maximum principles and the total variation diminishing (TVD) property. The study of these properties will be done within the framework of partitioned Runge-Kutta methods. It will also be seen that the incompatibility of consistency and mass-conservation holds for ‘genuine’ multirate schemes, but not for general partitioned methods.

This contribution discusses the construction of kernel-based adaptive particle methods for numerical flow simulation, where the finite volume particle method (FVPM) is used as a prototype. In the FVPM, scattered data approximation algorithms are required in the recovery step of the WENO reconstruction. We first show how kernel-based approximation schemes can be used in the recovery step of particle methods, where we give preference to the radial polyharmonic spline kernel. Then we discuss important aspects concerning the numerical stability and approximation behaviour of polyharmonic splines. Moreover, we propose customized coarsening and refinement rules for the adaptive resampling of the particles. Supporting numerical examples and comparisons with other radial kernels are provided.

Although high order discontinuous Galerkin spectral element methods (DGSEMs) formally require more work than high order finite difference methods, they are not necessarily more expensive in practice. The matrix-vector multiplication used to compute the spatial derivatives compares favorably to the work needed to solve the tri-diagonal systems required by compact finite difference schemes. The stiffness of the two approximations is not significantly different because of the stretching used in finite difference methods to reduce the accuracy loss near boundaries. Implicit methods can give speedups close to two orders of magnitude when memory and time accuracy are not issues. However for time accurate problems much of the advantage is lost because the increase in the time step need not be enough to counterbalance the increased work per time step. Parallelism complicates the situation because the explicit time integration of the DGSEM is effectively parallelized. Overall, the evidence suggests that implicit methods are not the panacea for reducing the stiffness barriers for high order DGSEMs.

Viscous hydrodynamic flow simulations in the framework of unstructured, interface capturing, finite-volume methods have matured to an industrial standard by now. Accordingly, the prediction of maritime free-surface flows around floating bodies is a standard application for naval architects. Typical examples refer to the drag of a ship in steady calm-water conditions, a vessel’s seaway performance or tank sloshing.

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.

Finite volume and discontinuous Galerkin methods are powerful computational tools for the solution of systems of conservation laws as the Navier Stokes equations. This is due to the fact that they allow piecewise continuous approximations, which turned out to be more robust especially in under-resolved regions or near shock waves. The idea of this paper is to apply an a posteriori post-processing of a steady state solution of a finite volume or a discontinuous Galerkin scheme. The approximation, which consists in every grid cell of a polynomial of degree N, is shifted to polynomials of degree M by reconstruction. The improved approximate solution is inserted into a higher-order approximation to estimate the local discretization error of the obtained solution. This estimated local discretization error of the basic scheme is subtracted from the right hand side of the basic scheme. A new steady state solution is calculated by the modified basic scheme. Iteratively applied, commutes the defect correction the approximation to a steady state solution of higher-order accuracy. For the correction one only needs the inversion of the basic lower-order scheme within an iteration loop. The modification of the basic scheme is non-intrusive and restricted to a change of the right hand side.

We discuss linear and nonlinear boundary conditions for wave propagation problems. The concepts of well-posedness and stability are discussed by considering a specific example of a boundary condition occurring in the modeling of earthquakes. That boundary condition can be formulated in a linear and nonlinear way and implemented in a characteristic and non-characteristic way. These differences are discussed and the implications and difficulties are pointed out. Numerical simulations that illustrate the theoretical discussion are presented together with an application that show that the methodology can be used for practical problems.

One possibility to solve stiff ODEs like the example of Prothero and Robinson [21] or differential algebraic equations are Runge-Kutta methods (RK methods) [9, 31]. Explicit RK methods may not be a good choice since for getting a stable numerical solution a stepsize restriction should be accepted, i.e. the problem should be solved with very small timesteps. Therefore it might be better to use implicit or linear implicit RK methods, so-called Rosenbrock–Wanner methods. Fully implicit RK methods may be ineffective for solving high dimensional ODEs since they need a high computational effort to solve the huge nonlinear system. Therefore we consider in this note diagonally implicit RK methods (DIRK methods).

We extend the Spectral Difference method to Proriol-Koornwinder-Dubi-ner-polynomials (PKD) on triangular grids using a two dimensional Lobatto points extension as the set of fluxpoints. These polynomials form an orthogonal basis on triangles and fulfill a singular Sturm-Liouville-problem which can be used to construct modal filters in order to stabilize the scheme for nonlinear conservation laws. To avoid global filtering, we give an outlook of possible edge detection techniques in two dimensions based on the conjugated Fourier partial sum. Finally, we show numerical results for the Spectral Difference method using the proposed filter technique applied to the Euler equations and the nonlinear shock vortex interaction.