查看更多>>摘要：This paper presents a mass and momentum conservative semi-implicit finite volume(FV)scheme for complex non-hydrostatic free surface flows,interacting with moving solid obstacles.A simplified incompressible Baer-Nunziato type model is considered for two-phase flows containing a liquid phase,a solid phase,and the surrounding void.According to the so-called diffuse interface approach,the different phases and consequently the void are described by means of a scalar volume fraction function for each phase.In our numeri-cal scheme,the dynamics of the liquid phase and the motion of the solid are decoupled.The solid is assumed to be a moving rigid body,whose motion is prescribed.Only after the advection of the solid volume fraction,the dynamics of the liquid phase is considered.As usual in semi-implicit schemes,we employ staggered Cartesian control volumes and treat the nonlinear convective terms explicitly,while the pressure terms are treated implicitly.The non-conservative products arising in the transport equation for the solid volume frac-tion are treated by a path-conservative approach.The resulting semi-implicit FV discretiza-tion of the mass and momentum equations leads to a mildly nonlinear system for the pres-sure which can be efficiently solved with a nested Newton-type technique.The time step size is only limited by the velocities of the two phases contained in the domain,and not by the gravity wave speed nor by the stiff algebraic relaxation source term,which requires an implicit discretization.The resulting semi-implicit algorithm is first validated on a set of classical incompressible Navier-Stokes test problems and later also adds a fixed and mov-ing solid phase.
查看更多>>摘要：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 frac-tion of the application mix on such supercomputers.It is,therefore,very important to pre-pare 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 exception-ally 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 algo-rithms 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 indistin-guishable 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 mimeti-cally 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 opti-mal stratagem for overcoming the higher computational complexities of higher order schemes.Several avenues for future improvement have also been identified.A scalability study is pre-sented for a real-world application using GPUs and comparable numbers of high-end mul-ticore 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.
查看更多>>摘要：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.
查看更多>>摘要：We consider the mixed discontinuous Galerkin(DG)finite element approximation of the Stokes equation and provide the analysis for the[Pk]d-Pk-1 element on the tensor product meshes.Comparing to the previous stability proof with[Qk]d-Qk-1 discontinuous finite elements in the existing references,our first contribution is to extend the formal proof to the[Pk]d-Pk-1 discontinuous elements on the tensor product meshes.Numerical inf-sup tests have been performed to compare Qk and Pk 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[Pk]d-Pk-1 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.
查看更多>>摘要：In this paper,we consider two Cauchy systems of coupled two wave equations in the whole line R 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 L2(R)-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 pre-sent 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 consid-ered 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.
查看更多>>摘要：In this paper,a two-step iteration method is established which can be viewed as a generali-zation 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.
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