查看更多>>摘要：Hyperbolic conservation laws arise in the context of continuum physics,and are mathemat-ically 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.
查看更多>>摘要：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.
查看更多>>摘要：A novel numerical scheme to solve two coupled systems of conservation laws is intro-duced.The scheme is derived based on a relaxation approach and does not require informa-tion on the Lax curves of the coupled systems,which simplifies the computation of suit-able 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 prob-lem 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.
查看更多>>摘要：This paper presents a high-order discontinuous Galerkin(DG)finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynami-cally compatible(SHTC)model of compressible two-phase flow,introduced by Romenski et al.in[59,62],in multiple space dimensions.In the absence of algebraic source terms,the model is endowed with a curl constraint on the relative velocity field.In this paper,the hyperbolicity of the system is studied for the first time in the multidimensional case,show-ing that the original model is only weakly hyperbolic in multiple space dimensions.To restore the strong hyperbolicity,two different methodologies are used:(ⅰ)the explicit sym-metrization of the system,which can be achieved by adding terms that contain linear com-binations of the curl involution,similar to the Godunov-Powell terms in the MHD equa-tions;(ⅱ)the use of the hyperbolic generalized Lagrangian multiplier(GLM)curl-cleaning approach forwarded.The PDE system is solved using a high-order ADER-DG method with a posteriori subcell finite-volume limiter to deal with shock waves and the steep gradients in the volume fraction commonly appearing in the solutions of this type of model.To illus-trate the performance of the method,several different test cases and benchmark problems have been run,showing the high order of the scheme and the good agreement when com-pared to reference solutions computed with other well-known methods.
查看更多>>摘要：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 dimen-sionally 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 split-ting 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 projec-tions with weak enforcement of the boundary conditions at the bottom and at the free sur-face.The resulting equation systems of order zero and one are presented in linear and non-linear 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.
查看更多>>摘要：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 clas-sify 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 clas-sified and confirm whether sub-shocks emerge.
查看更多>>摘要：In this paper,we study the convergence of a second-order finite volume approximation of the scalar conservation law.This scheme is based on the generalized Riemann problem(GRP)solver.We first investigate the stability of the GRP scheme and find that it might be entropy-unstable when the shock wave is generated.By adding an artificial viscosity,we propose a new stabilized GRP scheme.Under the assumption that numerical solutions are uniformly bounded,we prove the consistency and convergence of this new GRP method.
查看更多>>摘要：The solution of time-dependent hyperbolic conservation laws on cut cell meshes causes the small cell problem:standard schemes are not stable on the arbitrarily small cut cells if an explicit time stepping scheme is used and the time step size is chosen based on the size of the background cells.In May and Berger(J Sci Comput 71:919-943,2017),the mixed explicit-implicit approach in general and MUSCL-Trap(monotonic upwind scheme for con-servation laws and trapezoidal scheme)in particular have been introduced to solve this prob-lem by using implicit time stepping on the cut cells.Theoretical and numerical results have indicated that this might lead to a loss in accuracy when switching between the explicit and implicit time stepping.In this contribution,we examine this in more detail and will prove in one dimension that the specific combination MUSCL-Trap of an explicit second-order and an implicit second-order scheme results in a fully second-order mixed scheme.As this result is unlikely to hold in two dimensions,we also introduce two new versions of mixed explicit-implicit schemes based on exchanging the explicit scheme.We present numerical tests in two dimensions where we compare the new versions with the original MUSCL-Trap scheme.
查看更多>>摘要：Understanding the dynamics of phase boundaries in fluids requires quantitative knowledge about the microscale processes at the interface.We consider the sharp-interface motion of the compressible two-component flow and propose a heterogeneous multiscale method(HMM)to describe the flow fields accurately.The multiscale approach combines a hyper-bolic system of balance laws on the continuum scale with molecular-dynamics(MD)simu-lations on the microscale level.Notably,the multiscale approach is necessary to compute the interface dynamics because there is—at present—no closed continuum-scale model.The basic HMM relies on a moving-mesh finite-volume method and has been introduced recently for the compressible one-component flow with phase transitions by Magiera and Rohde in(J Comput Phys 469:111551,2022).To overcome the numerical complexity of the MD microscale model,a deep neural network is employed as an efficient surrogate model.The entire approach is finally applied to simulate droplet dynamics for argon-methane mixtures in several space dimensions.To our knowledge,such compressible two-phase dynamics accounting for microscale phase-change transfer rates have not yet been computed.
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