Hyperbolic conservation laws arise in the context of continuum physics, and are mathematically presented in differential form and understood in the distributional (weak) sense. The formal application of the Gauss-Green theorem results in integral balance laws, in which the concept of flux plays a central role. This paper addresses the spacetime viewpoint of the flux regularity, providing a rigorous treatment of integral balance laws. The established Lipschitz regularity of fluxes (over time intervals) leads to a consistent flux approximation. Thus, fully discrete finite volume schemes of high order may be consistently justified with reference to the spacetime integral balance laws.
Transpiration cooling is numerically investigated, where a cooling gas is injected through a carbon composite material into a hot gas channel. To account for microscale effects at the injection interface, an effective problem is derived. Here, effects induced by microscale structures on macroscale variables, e.g., cooling efficiency, are taken into account without resolving the microscale structures. For this purpose, effective boundary conditions at the interface between hot gas and porous medium flow are derived using an upscaling strategy. Numerical simulations in 2D with effective boundary conditions are compared to uniform and non-uniform injection. The computations confirm that the effective model provides a more efficient and accurate approximation of the cooling efficiency than the uniform injection.
A novel numerical scheme to solve two coupled systems of conservation laws is introduced. The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems, which simplifies the computation of suitable coupling data. The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit. The role of this condition is discussed, a consistency concept with respect to the original problem is introduced, the well-posedness is analyzed and explicit, nodal Riemann solvers are provided. Based on a case study considering the p-system of gas dynamics, a strategy for the design of the relaxation coupling condition within the new scheme is provided.
This paper presents a high-order discontinuous Galerkin (DG) finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) model of compressible two-phase flow, introduced by Romenski et al. in [
Shallow moment models are extensions of the hyperbolic shallow water equations. They admit variations in the vertical profile of the horizontal velocity. This paper introduces a non-hydrostatic pressure to this framework and shows the systematic derivation of dimensionally reduced dispersive equation systems which still hold information on the vertical profiles of the flow variables. The derivation from a set of balance laws is based on a splitting of the pressure followed by a same-degree polynomial expansion of the velocity and pressure fields in a vertical direction. Dimensional reduction is done via Galerkin projections with weak enforcement of the boundary conditions at the bottom and at the free surface. The resulting equation systems of order zero and one are presented in linear and nonlinear forms for Legendre basis functions and an analysis of dispersive properties is given. A numerical experiment shows convergence towards the resolved reference model in the linear stationary case and demonstrates the reconstruction of vertical profiles.
We study the shock structure and the sub-shock formation in a binary mixture of rarefied polyatomic gases, considering the dissipation only due to the dynamic pressure. We classify the regions depending on the concentration and the Mach number for which there may exist the sub-shock in the profile of shock structure in one or both constituents or not for prescribed values of the mass ratio of the constituents and the ratios of the specific heats. We compare the regions with the ones of the corresponding mixture of Eulerian gases and perform the numerical calculations of the shock structure for typical cases previously classified and confirm whether sub-shocks emerge.
GPU computing is expected to play an integral part in all modern Exascale supercomputers. It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such supercomputers. It is, therefore, very important to prepare the community of users of higher order schemes for hyperbolic PDEs for this emerging opportunity.
Not every algorithm that is used in the space-time update of the solution of hyperbolic PDEs will take well to GPUs. However, we identify a small core of algorithms that take exceptionally well to GPU computing. Based on an analysis of available options, we have been able to identify weighted essentially non-oscillatory (WENO) algorithms for spatial reconstruction along with arbitrary derivative (ADER) algorithms for time extension followed by a corrector step as the winning three-part algorithmic combination. Even when a winning subset of algorithms has been identified, it is not clear that they will port seamlessly to GPUs. The low data throughput between CPU and GPU, as well as the very small cache sizes on modern GPUs, implies that we have to think through all aspects of the task of porting an application to GPUs. For that reason, this paper identifies the techniques and tricks needed for making a successful port of this very useful class of higher order algorithms to GPUs.
Application codes face a further challenge—the GPU results need to be practically indistinguishable from the CPU results—in order for the legacy knowledge bases embedded in these applications codes to be preserved during the port of GPUs. This requirement often makes a complete code rewrite impossible. For that reason, it is safest to use an approach based on OpenACC directives, so that most of the code remains intact (as long as it was originally well-written). This paper is intended to be a one-stop shop for anyone seeking to make an OpenACC-based port of a higher order Godunov scheme to GPUs.
We focus on three broad and high-impact areas where higher order Godunov schemes are used. The first area is computational fluid dynamics (CFD). The second is computational magnetohydrodynamics (MHD) which has an involution constraint that has to be mimetically preserved. The third is computational electrodynamics (CED) which has involution constraints and also extremely stiff source terms. Together, these three diverse uses of higher order Godunov methodology, cover many of the most important applications areas. In all three cases, we show that the optimal use of algorithms, techniques, and tricks, along with the use of OpenACC, yields superlative speedups on GPUs. As a bonus, we find a most remarkable and desirable result: some higher order schemes, with their larger operations count per zone, show better speedup than lower order schemes on GPUs. In other words, the GPU is an optimal stratagem for overcoming the higher computational complexities of higher order schemes. Several avenues for future improvement have also been identified. A scalability study is presented for a real-world application using GPUs and comparable numbers of high-end multicore CPUs. It is found that GPUs offer a substantial performance benefit over comparable number of CPUs, especially when all the methods designed in this paper are used.
Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely, to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.
We consider the mixed discontinuous Galerkin (DG) finite element approximation of the Stokes equation and provide the analysis for the element on the tensor product meshes. Comparing to the previous stability proof with discontinuous finite elements in the existing references, our first contribution is to extend the formal proof to the discontinuous elements on the tensor product meshes. Numerical inf-sup tests have been performed to compare and types of elements and validate the well-posedness in both settings. Secondly, our contribution is to design the enhanced pressure-robust discretization by only modifying the body source assembling on schemes to improve the numerical simulation further. The produced numerical velocity solution via our enhancement shows viscosity and pressure independence and thus outperforms the solution produced by standard discontinuous Galerkin schemes. Robustness analysis and numerical tests have been provided to validate the scheme’s robustness.
In this paper, we consider two Cauchy systems of coupled two wave equations in the whole line under one or two frictional dampings, where the coupling terms are either of order one with respect to the time variable or of order two with respect to the space variable. We prove some -norm decay estimates of solutions and their higher-order derivatives with respect to the space variable, where the decay rates depend on the number of the present frictional dampings, the regularity of the initial data, and some connections between the speeds of wave propagation of the two wave equations. Both our systems are considered under weaker conditions on the coefficients than the ones considered in the literature and they include the case where only one frictional damping is present, so they generate new difficulties and represent new situations that have not been studied earlier.
In this paper, a two-step iteration method is established which can be viewed as a generalization of the existing modulus-based methods for vertical linear complementarity problems given by He and Vong (Appl. Math. Lett. 134:108344, 2022). The convergence analysis of the proposed method is established, which can improve the existing results. Numerical examples show that the proposed method is efficient with the two-step technique.