On structure preserving numerical schemes for hyperbolic partial differential equations and multiscale kinetic equations

Abstract

Natural phenomena are frequently represented through the formulation of differential equations, coupled with specific initial and boundary conditions. Many such models possess inherent structures that are crucial in describing the behaviour of solutions. Unfortunately, numerical discretisations of such models often fail to preserve these structures, leading to inappropriate numerical solutions. The numerical schemes that take special care to preserve the inherent structures of a given differential equation in its discretisation process are known as structure preserving schemes. Various structures have been extensively discussed in the existing literature. This thesis focuses on crucial structure preserving strategies such as entropy stability, asymptotic preservation and well-balancing. Many hyperbolic systems of partial differential equations (PDEs) have entropy inequalities associated with them. Numerical schemes that are designed to inherently satisfy the entropy inequality are known as entropy stable schemes. On the other hand, the hyperbolic system of PDEs can be derived as an approximation of a vector-kinetic model, which also encompasses associated entropy structures. The entropy inequality of the hyperbolic system of PDE can be derived as a moment of the entropy structures of vector-kinetic model. However, this correspondence is not maintained in numerical discretisations. Presented as the first part of this thesis is the development and analysis of a numerical scheme that achieves entropy stability for the vector-kinetic model, along with the proof that it also recovers entropy stability for the given hyperbolic system of PDEs. Hyperbolic and kinetic equations containing small spatial and temporal scales due to stiff source terms or strong forcing, pose significant challenges for numerical approximation. Asymptotic preserving (AP) schemes offer an effective solution for handling these asymptotic regimes, allowing for efficient computations without the need for excessively small mesh sizes and time steps. Unlike traditional domain decomposition methods that involve coupling different models (in different regimes) through interface conditions, AP schemes seamlessly transition between different scales by ensuring automatic adaptation of solvers based on the resolution of scales. Presented as the second part of this thesis is the development and analysis of a high order AP scheme for diffusive-scaled linear kinetic equations with general initial conditions. The dimensionless form of barotropic Euler system contains the parameter Mach number which can become small, and this results in the need for an AP scheme. Moreover, this system has an entropy inequality corresponding to a convex entropy function, for all values of the parameter. Hence, this system requires treatment with regard to both the structures: asymptotic preservation and entropy stability. Presented as a third part of this thesis is the development and analysis of an AP scheme satisfying entropy stability for all values of the parameter in the barotropic Euler system. In the fourth part of this thesis, the mathematical properties of Lattice Boltzmann Methods (LBMs) derived from vector-kinetic models of hyperbolic PDEs are presented. This LBM framework is extended to hyperbolic PDEs with stiff source terms, where suitable modeling at the vector-kinetic level combined with well-balancing is introduced to avoid spurious numerical convection arising from the discretisation of source terms and thereby avoiding wave propagation at incorrect speeds.