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Register a new userFinite volume schemes and Lax-Wendroff consistency
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Added by
Raphaele Herbin
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We present a (partial) historical summary of the mathematical analysis of finite differences and finite volumes methods, paying a special attention to the Lax-Richtmyer and Lax-Wendroff theorems. We then state a Lax-Wendroff consistency result for convection operators on staggered grids (often used in fluid flow simulations), which illustrates a recent generalization of the flux consistency notion designed to cope with general discrete functions.
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This paper addresses the three concepts of textit{ consistency, stability and convergence } in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of balance laws. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of Godunov compatibility, which serves as a substitute to the entropy condition.
In this paper, we present a class of finite volume schemes for incompressible flow problems. The unknowns are collocated at the center of the control volumes, and the stability of the schemes is obtained by adding to the mass balance stabilization terms involving the pressure jumps across the edges of the mesh.
The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a black-box solver. The Non Intrusive Reduced Basis method (NIRB) has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finite volume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meenings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to Finite Volume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid-Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA).
We are interested in simulating blood flow in arteries with variable elasticity with a one dimensional model. We present a well-balanced finite volume scheme based on the recent developments in shallow water equations context. We thus get a mass conservative scheme which also preserves equilibria of Q=0. This numerical method is tested on analytical tests.
A time-fractional Allen-Cahn equation with volume constraint is first proposed by introducing a nonlocal time-dependent Lagrange multiplier. Adaptive linear second-order energy stable schemes are developed for the proposed model by combining invariant energy quadratization and scalar auxiliary variable approaches with the recent L1$^{+}$ formula. The new developed methods are proved to be volume-preserving and unconditionally energy stable on arbitrary nonuniform time meshes. The accelerated algorithm and adaptive time strategy are employed in numerical implement. Numerical results show that the proposed algorithms are computationally efficient in multi-scale simulations, and appropriate for accurately resolving the intrinsically initial singularity of solution and for efficiently capturing the fast dynamics away initial time.
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