Towards a New-generation Numerical Scheme for the Compressible Navier-Stokes Equations With the Active Flux Method

Abstract

Taking underlying physics into consideration is essential in the design of numerical schemes for fluid dynamics simulations. The Active Flux (AF) method is a novel class of fully-discrete numerical schemes that aim to truly reflect the multidimensional physics in gas dynamics. The original scheme has successfully achieved third-order accuracy for the Euler equations. In this project, we overhaul the original scheme and formulate a general paradigm for the Active Flux method, which allows a flexible and systematic ex- tension to a family of schemes as well as a tidy treatment of general boundary conditions. We propose a novel and versatile notion of discrete conservation laws for the general AF method. We also apply the schemes to solving the Navier-Stokes equations. Numerical strategies for approximating derivatives and improving stability are discussed along the way, which significantly expands the capacity of the AF method. Throughout the dissertation, we view the AF method as a predictor-corrector method. In the original AF scheme, assigning degrees of freedom to the cell boundaries and reconstructing continuous piecewise functions are key to capture the multidimensional physics in the inviscid flows at the prediction stage. However, having degrees of freedom (DOFs) shared by neighboring cells poses difficulty in enforcing conservation laws in the correction stage. We introduce a novel treatment of discrete conservation laws for the AF method. Instead of enforcing strict local conservation for each cell, we dis- tribute to the nodal values a “discrepancy” equal to the difference between conservative and non-conservative updates, which replaces expensive and inconsistent bubble functions that were used in the original 3rd-order AF scheme. This notion of conservation is compatible with a family of arbitrary-order spatial discretization and independent of the predictor. Thus we are able to systematically extend the AF method to a family of schemes of various order of accuracy and apply them to different problems governed by conservation laws. Analysis in one space dimension and abundant test cases, including the Euler equations, in both one and two space dimensions, are presented to verify the accuracy. When we extend the Euler solver to the Navier-Stokes equations, we face difficulty from the stiffness caused by viscosity and heat conduction as well as approximating second-order derivatives. We introduce relaxed variables to reformulate the equations as a first-order hyperbolic system. The hyperbolic model has the same steady-state solutions as the original one. Our numerical experiments indicate that, compared with the benchmark scheme that directly approximates second-order derivatives, using the hyperbolic reformulation in solving the steady-state problems significantly reduces the simulation time while yielding better accuracy. In practice, most problems in fluid dynamics simulations come with domains that contain complex geometry. Within the framework that we propose for the AF method, we discuss a simple but effective implementation of general boundary conditions in predictors and correctors. Results generated by the second-order scheme from the AF family are presented for both inviscid and viscous flows on domains with curved boundaries.