Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)

This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well

Maximum principle preserving time implicit DGSEM for linear scalar hyperbolic conservation laws

We investigate the properties of the high-order discontinuous Galerkin spectral element method (DGSEM) with implicit backward-Euler time stepping for the approximation of hyperbolic linear scalar conservation equation in multiple space dimensions. We first prove that the DGSEM scheme in one space dimension preserves a maximum principle for the cell-averaged solution when the time step is large enough. This property however no longer holds in multiple space dimensions and we propose to use the flux-corrected transport limiting [Boris and Book, J. Comput. Phys., 11 (1973)] based on a low-order approximation using graph viscosity to impose a maximum principle on the cell-averaged solution. These results allow to use a linear scaling limiter [Zhang and Shu, J. Comput. Phys., 229 (2010)] in order to impose a maximum principle at nodal values within elements. Then, we investigate the inversion of the linear systems resulting from the time implicit discretization at each time step. We prove that the diagonal blocks are invertible and provide efficient algorithms for their inversion. Numerical experiments in one and two space dimensions are presented to illustrate the conclusions of the present analyses.Comment: 34 page

A study on CFL conditions for the DG solution of conservation laws on adaptive moving meshes

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step has been based on CFL conditions established for fixed and uniform meshes. This work provides a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the $\alpha$-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the $\alpha$-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented.Comment: 21 page

Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations

In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations