GPU Accelerated Explicit Time Integration Methods for Electro-Quasistatic Fields

Electro-quasistatic field problems involving nonlinear materials are commonly discretized in space using finite elements. In this paper, it is proposed to solve the resulting system of ordinary differential equations by an explicit Runge-Kutta-Chebyshev time-integration scheme. This mitigates the need for Newton-Raphson iterations, as they are necessary within fully implicit time integration schemes. However, the electro-quasistatic system of ordinary differential equations has a Laplace-type mass matrix such that parts of the explicit time-integration scheme remain implicit. An iterative solver with constant preconditioner is shown to efficiently solve the resulting multiple right-hand side problem. This approach allows an efficient parallel implementation on a system featuring multiple graphic processing units.Comment: 4 pages, 5 figure

Monolithic multigrid methods for high-order discretizations of time-dependent PDEs

A currently growing interest is seen in developing solvers that couple high-fidelity and higher-order spatial discretization schemes with higher-order time stepping methods for various time-dependent fluid plasma models. These problems are famously known to be stiff, thus only implicit time-stepping schemes with certain stability properties can be used. Of the most powerful choices are the implicit Runge-Kutta methods (IRK). However, they are multi-stage, often producing a very large and nonsymmetric system of equations that needs to be solved at each time step. There have been recent efforts on developing efficient and robust solvers for these systems. We have accomplished this by using a Newton-Krylov-multigrid approach that applies a multigrid preconditioner monolithically, preserving the system couplings, and uses Newton’s method for linearization wherever necessary. We show robustness of our solver on the single-fluid magnetohydrodynamic (MHD) model, along with the (Navier-)Stokes and Maxwell’s equations. For all these, we couple IRK with higher-order (mixed) finiteelement (FEM) spatial discretizations. In the Navier-Stokes problem, we further explore achieving more higher-order approximations by using nonconforming mixed FEM spaces with added penalty terms for stability. While in the Maxwell problem, we focus on the rarely used E-B form, where both electric and magnetic fields are differentiated in time, and overcome the difficulty of using FEM on curved domains by using an elasticity solve on each level in the non-nested hierarchy of meshes in the multigrid method

Efficient Solvers for Space-Time Discontinuous Galerkin Spectral Element Methods

In this thesis we study efficient solvers for space-time discontinuous Galerkin spectral element methods (DG-SEM). These discretizations result in fully implicit schemes of variable order in both spatial and temporal directions. The popularity of space-time DG methods has increased in recent years and entropy stable space-time DG-SEM have been constructed for conservation laws, making them interesting for these applications. The size of the nonlinear system resulting from differential equations discretized with space-time DG-SEM is dependent on the order of the method, and the corresponding Jacobian is of block form with dense blocks. Thus, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU time as well as memory consumption. The lack of good solvers for three-dimensional DG applications has been identified as one of the major obstacles before high order methods can be adapted for industrial applications.It has been proven that DG-SEM in time and Lobatto IIIC Runge-Kutta methods are equivalent, in that both methods lead to the same discrete solution. This allows to implement space-time DG-SEM in two ways: Either as a full space-time system or by decoupling the temporal elements and using implicit time-stepping with Lobatto IIIC methods. We compare theoretical properties and discuss practical aspects of the respective implementations.When considering the full space-time system, multigrid can be used as solver. We analyze this solver with the local Fourier analysis, which gives more insight into the efficiency of the space-time multigrid method. The other option is to decouple the temporal elements and use implicit Runge-Kutta time-stepping methods. We suggest to use Jacobian-free Newton-Krylov (JFNK) solvers since they are advantageous memory-wise. An efficient preconditioner for the Krylov sub-solver is needed to improve the convergence speed. However, we want to avoid constructing or storing the Jacobian, otherwise the favorable memory consumption of the JFNK approach would be obsolete. We present a preconditioner based on an auxiliary first order finite volume replacement operator. Based on the replacement operator we construct an agglomeration multigrid preconditioner with efficient smoothers using pseudo time integrators. Then only the Jacobian of the replacement operator needs to be constructed and the DG method is still Jacobian-free. Numerical experiments for hyperbolic test problems as the advection, advection-diffusion and Euler equations in several dimensions demonstrate the potential of the new approach

Parallel-in-Time Solver for the All-at-Once Runge--Kutta Discretization

In this article, we derive fast and robust parallel-in-time preconditioned iterative methods for the all-at-once linear systems arising upon discretization of time-dependent PDEs. The discretization we employ is based on a Runge--Kutta method in time, for which the development of parallel solvers is an emerging research area in the literature of numerical methods for time-dependent PDEs. By making use of classical theory of block matrices, one is able to derive a preconditioner for the systems considered. The block structure of the preconditioner allows for parallelism in the time variable, as long as one is able to provide an optimal solver for the system of the stages of the method. We thus propose a preconditioner for the latter system based on a singular value decomposition (SVD) of the (real) Runge--Kutta matrix $A_{\mathrm{RK}} = U \Sigma V^\top$. Supposing $A_{\mathrm{RK}}$ is invertible, we prove that the spectrum of the system for the stages preconditioned by our SVD-based preconditioner is contained within the right-half of the unit circle, under suitable assumptions on the matrix $U^\top V$ (the assumptions are well posed due to the polar decomposition of $A_{\mathrm{RK}}$). We show the numerical efficiency of our SVD-based preconditioner by solving the system of the stages arising from the discretization of the heat equation and the Stokes equations, with sequential time-stepping. Finally, we provide numerical results of the all-at-once approach for both problems, showing the speed-up achieved on a parallel architecture

A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure

In this paper we formulate and test numerically a fully-coupled discontinuous Galerkin (DG) method for incompressible two-phase flow with discontinuous capillary pressure. The spatial discretization uses the symmetric interior penalty DG formulation with weighted averages and is based on a wetting-phase potential / capillary potential formulation of the two-phase flow system. After discretizing in time with diagonally implicit Runge-Kutta schemes the resulting systems of nonlinear algebraic equations are solved with Newton's method and the arising systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method. The new scheme is investigated for various test problems from the literature and is also compared to a cell-centered finite volume scheme in terms of accuracy and time to solution. We find that the method is accurate, robust and efficient. In particular no post-processing of the DG velocity field is necessary in contrast to results reported by several authors for decoupled schemes. Moreover, the solver scales well in parallel and three-dimensional problems with up to nearly 100 million degrees of freedom per time step have been computed on 1000 processors

Explicit stabilized integrators for stiff optimal control problems

Explicit stabilized methods are an efficient alternative to implicit schemes for the time integration of stiff systems of differential equations in large dimension. In this paper, we derive explicit stabilized integrators of orders one and two for the optimal control of stiff systems. We analyze their favorable stability properties based on the continuous optimality conditions. Furthermore, we study their order of convergence taking advantage of the symplecticity of the corresponding partitioned Runge-Kutta method involved for the adjoint equations. Numerical experiments including the optimal control of a nonlinear diffusion-advection PDE illustrate the efficiency of the new approach.Comment: 23 page