Internal Research Project I-07

Review of certain grid refinement methods for the Navier-Stokes equations


September 2022 - August 2023


About this project

Coupled free-flow porous-medium-flow systems are the focus of project area A of SFB1313. Particularly for a rough surface topology, flow patterns in free flow become complicated. This motivates the investigation of methods that allow to resolve details of those complex flow patterns.
We describe the free flow by the Navier-Stokes equations. To admit local mass and momentum conservation, we focus on finite-volume methods, which are often equivalent to finite-difference methods in this context. Although we also examine degree-of-freedom arrangments other than marker-and-cell, our main focus is on the marker-and-cell method, which is used frequently and which avoids spurious pressure oscillations. For simplicity, we will also focus on methods with rectangular control volumes.

We will first review the variants that are possible for such spatial discretization methods. This involves types of grids that can be used, with a focus on leaf, level and overlapping grids. It further includes to look at possbile arrangements of computational variables such as the different Arakawa grids as well as different arrangements
at refinement interfaces. We will review arrangments of control volumes including overlapping arrangments, covering ones and ones with gaps. Then the discretization of balance equations involving the concept of virtual
variables, stencils and cells should be looked at. Further, options for the interpolation or extrapolation of virtual variables will be reviewed in detail, discussing also conservative and nonconservative methods as well as restriction and prolongation operators. Relations between fine and coarse quantities at refinement interfaces and in overlapping zones will be discussed as well as aspects of solution procedures in this context, especially multigrid methods.

We then will review the state of the art of the theoretical and numerical investigation of such finite-volume and finite-difference methods, which includes as a major aspect truncation errors, how they are influenced by interpolations and how they are related to superconvergence effects. Other major aspects are the convergence rates in different error norms as well as the comparison of errors for different grids with the same number of degrees of freedom. As minor aspects we will discuss literature on wiggles occuring at refinement interfaces as well as the symmtery of Poisson matrices.

Next, we will give own ideas on alternative methods with a focus on the issue of conservation and order of the local truncation errors. We will also expand the theoretical and numerical investigation of those numerical methods for the Naviers-Stokes equations. The further theoretical investigation will among others further tackle the issues of the truncation errors, the relationship between different methods and the influence of distorted stencils. The numerical investigation will emphasize on the relationsship between the local distribution of truncation errors and the local distribution of solution errors. For this, quadtree marker-and-cell methods already implemented in the open-source software Dumux will be used. Further, a comparison to external software, namely IBAMR, will be included.

With this thorough investigation of the free-flow discretization at hand, we will apply refinement of the freeflow-regime to coupled porous-medium free-flow problems. To not overload one publication, we plan to prepare a first publication of the pure free-flow methods focussing on literature review. The application to porous media will be included after that.

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