Efficient Parallel AMG Methods For Discrete Systems Of Some Partial Differential Equations

Algebraic multigrid(AMG)is a popular method for solving large-scale discrete systems of partial differential equations(PDEs),and its parallel AMG solver is the preferred solver for many commercial computing software platforms.Although the AMG method has been relatively mature after more than 40 years of development,with the deepening and development of scientific computing,various complex PDEs models continue to emerge.Therefore,it is still necessary to study the AMG method with universality,usability,low computational complexity and good scalability.Based on the characteristics of the discrete schemes for several types of PDEs with important application background,this paper focuses on the efficient(parallel)AMG methods for solving their discrete algebraic systems.The main work is as follows:Aiming at the three-temperature radiation diffusion equations with complicated nonlinear couplings among their physical quantities at multiple temporal and spatial scales,the parallel AMG method of its quite ill-conditioned fully coupled implicit symmetric-preserving finite volume element discrete system is studied.A new upper and lower block triangular preconditioners based on the AMG method are designed,and prove strictly that the spectral radius of these two preconditioned matrices are independent of the discrete problem size.The results of serial and parallel numerical comparison experiments show that compared with the commonly used AMG preconditioners,the Bramble-Pasciak CG method based on the lower triangle preconditioner and the GMRES(30)method based on the upper have good robustness and strong/weak scalability.For the Streamline-Upwind Petrov/Galerkin(SUPG)discrete system of the convection-diffusion equations,the adaptability of the classic AMG(CAMG)and the nonsymmetric AMG based on local approximate ideal restriction(?AIR)are first investigated;In order to improve their comprehensive performance,an adaptive AMG method based on the CAMG and the ?AIR is designed by introducing indicators that can characterize the asymmetric strength of the coefficient matrix,etc.The experimental results show that the adaptive AMG method has better convergence and higher computational efficiency.For the heat conduction equations and 2-T radiation diffusion equations on general polyhedral meshes,the parameter-containing nonlinear positivitypreserving finite volume(PPFV)schemes and the corresponding fast algorithm are studied.For the heat conduction equations,we first introduce a new parameter collocation method of one-sided flux,and further establish a parameter-containing nonlinear PPFV scheme based on two-point flux.Then,with the help of the Patankar trick,a novel and more concise nonlinear PPFV scheme with combined parameters is constructed.The selection strategy of corresponding collocation parameters and combination parameters in the above nonlinear schemes are also provided,and the existence and positivity-preserving properties of these two nonlinear PPFV solutions are proved.Since the above two nonlinear PPFV schemes are not necessary to assume that the values of the used auxiliary unknowns must be nonnegative,its interpolation formulae can be constructed by certain existing approaches with high accuracy and well robustness(e.g.,the finite element method).Numerical experiments under a variety of complex meshes are given to confirm our theoretical results,and the PPFV schemes can improve the convergence behavior of the Picard procedure in many cases.On the basis of the above work,a nonlinear PPFV scheme is designed for the 2-T radiation diffusion equations,and the positivity-preserving and energy conservation properties of the scheme are verified by numerical experiments.In addition,the influence of the above PPFV schemes on AMG are investigated,and the experiments show that the PPFV schemes can further improve the robustness and computational efficiency of AMG.