Speeding up high-order algorithms in computational fluid and kinetic dynamics: based on characteristics tracing and low-rank structures
Date
2023
Authors
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Publisher
University of Delaware
Abstract
Many physical phenomena can be described by nonlinear partial differential equations (PDEs). Yet, analytic solutions are oftentimes unavailable, and lab experiments can be time consuming and expensive; thus motivating the need for numerical solutions. Constructing low-storage, efficient and robust algorithms for solving PDEs comes with several computational challenges. First, classical discretization methods suffer from the curse of dimensionality, that is, the computational cost grows exponentially as the number of dimensions increases. Second, shock formations and sharp gradient structures that develop in certain PDEs of interest are challenging to capture due to the Gibbs phenomenon, that is, steep oscillations that occur near discontinuities. And third, satisfying physical properties such as conservation, equilibrium preservation and relative entropy dissipation is desired at the discrete level to avoid nonphysical behaviors. The goal of this dissertation is to develop efficient and robust algorithms for solving high-dimensional PDEs in fluid and kinetic applications. ☐ The first contribution of this dissertation is the development of a new Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for solving convection and convection-diffusion problems. Eulerian-Lagrangian and semi-Lagrangian methods have become popular ways to solve hyperbolic conservation laws due to their ability to allow large time-stepping sizes. The proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time-differential form to allow application of Runge-Kutta methods via the method of lines. The scheme can be viewed as a generalization of the standard Runge-Kutta finite volume (RK-FV) scheme for which the space-time region is partitioned by approximate characteristics with zero velocity. The high-order spatial reconstruction is achieved using the recently developed weighted essentially non-oscillatory scheme with adaptive order (WENO-AO); and the high-order temporal accuracy is achieved by explicit Runge-Kutta methods for convection equations and implicit-explicit (IMEX) Runge-Kutta methods for convection-diffusion equations. The algorithm extends to higher dimensions via dimensional splitting. Numerical experiments demonstrate the algorithm's robustness, high-order accuracy, and ability to handle extra large time-steps. ☐ The second contribution of this dissertation is the development of an implicit low-rank method for solving diffusion equations. Low-rank tensor methods have become a popular way to efficiently solve and store solutions of high-dimensional time-dependent PDEs. By taking advantage of low-rank structures inherent to some solutions of time-dependent problems, low-rank tensor methods reduce the storage requirements and hence the computational complexities. This helps avoid the curse of dimensionality. However, some PDEs of interest contain stiff operators that require implicit time integrators for reasonable computational efficiency. The proposed scheme is formulated by using traditional implicit time integrators to evolve the solution; decomposing the solution into one-dimensional time-dependent bases connected by time-dependent coefficients; and updating the one-dimensional bases one target direction at a time by freezing the solution in all other non-target directions. Projecting the equation onto the updated bases, the time-dependent coefficients are then updated. The updated solution is truncated using a basis removal procedure based on the singular value decomposition. The backward Euler method is used for the first-order scheme. Second-order schemes are also presented using second-order stiffly-accurate diagonally implicit Runge-Kutta methods, Crank-Nicolson method, and second-order backward differentiation formula. Numerical experiments demonstrate the algorithm's convergence and computational efficiency from enforcing the low-rank structure in solutions. ☐ The third contribution of this dissertation is the developement of a low-rank tensor method for solving the 1D2V Vlasov-Fokker-Planck (VFP) equation. The Vlasov-Fokker-Planck and Fokker-Planck type equations are kinetic models that are used to describe weakly coupled collisional plasmas. Developing efficient numerical methods for solving such models is of growing interest due to their applications in next-generation designs of field reversed configuration (FRC) thrusters and intertial confinement fusion (ICF) capsules. We consider a hybrid kinetic-ion fluid-electron model in which the ions are kinetically treated using the VFP equation and the electrons are treated using a fluid model. The proposed scheme is formulated by assuming a low-rank tensor structure of the solution in velocity space; discretizing the collision operator with the robust structure-preserving Chang-Cooper (SPCC) method; updating the solution by solving linear systems of tensor product structure; and truncating the solution using a basis removal procedure based on the singular value decomposition. Numerical experiments demonstrate the scheme's structure-preserving qualities, robustness and computational efficiency from enforcing the low-rank structure in solutions.
Description
Keywords
Robust algorithms, Partial differential equations, Implicit-explicit, Vlasov-Fokker-Planck equation