Towards learning Lattice Boltzmann collision operators

In this work we explore the possibility of learning from data collision operators for the Lattice Boltzmann Method using a deep learning approach. We compare a hierarchy of designs of the neural network (NN) collision operator and evaluate the performance of the resulting LBM method in reproducing time dynamics of several canonical flows. In the current study, as a first attempt to address the learning problem, the data was generated by a single relaxation time BGK operator. We demonstrate that vanilla NN architecture has very limited accuracy. On the other hand, by embedding physical properties, such as conservation laws and symmetries, it is possible to dramatically increase the accuracy by several orders of magnitude and correctly reproduce the short and long time dynamics of standard fluid flows

Mimetic finite difference method for the stokes problem on polygonal meshes

Various approaches to extend the finite element methods to non-traditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with low-order finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the second-order convergence for the velocity variable and the first-order for the pressure

Modeling anisotropic diffusion using a departure from isotropy approach

There are a large number of finite volume solvers available for solution of isotropic diffusion equation. This article presents an approach of adapting these solvers to solve anisotropic diffusion equations. The formulation works by decomposing the diffusive flux into a component associated with isotropic diffusion and another component associated with departure from isotropic diffusion. This results in an isotropic diffusion equation with additional terms to account for the anisotropic effect. These additional terms are treated using a deferred correction approach and coupled via an iterative procedure. The presented approach is validated against various diffusion problems in anisotropic media with known analytical or numerical solutions. Although demonstrated for two-dimensional problems, extension of the present approach to three-dimensional problems is straight forward. Other than the finite volume method, this approach can be applied to any discretization method

Dispersion reducing methods for edge discretizations of the electric vector wave equation

We present a novel strategy for minimizing the numerical dispersion error in edge discretizations of the time-domain electric vector wave equation on square meshes based on the mimetic finite difference (MFD) method. We compare this strategy, called M-adaptation, to two other discretizations, also based on square meshes. One is the lowest order Nédélec edge element discretization. The other is a modified quadrature approach (GY-adaptation) proposed by Guddati and Yue for the acoustic wave equation in two dimensions. All three discrete methods use the same edge-based degrees of freedom, while the temporal discretization is performed using the standard explicit Leapfrog scheme. To obtain efficient and explicit time stepping methods, the three schemes are further mass lumped. We perform a dispersion and stability analysis for the presented schemes and compare all three methods in terms of their stability regions and phase error. Our results indicate that the method produced by GY-adaptation and the Nédélec method are both second order accurate for numerical dispersion, but differ in the order of their numerical anisotropy (fourth order, versus second order, respectively). The result of M-adaptation is a discretization that is fourth order accurate for numerical dispersion as well as numerical anisotropy. Numerical simulations are provided that illustrate the theoretical results.Keywords: Nédélec edge elements, Vector wave equation, M-adaptation, Maxwell’s equations, Anisotropy, Dispersio