Comparison of different nonlinear solvers for 2D time-implicit stellar hydrodynamics

Time-implicit schemes are attractive since they allow numerical time steps that are much larger than those permitted by the Courant-Friedrich-Lewy criterion characterizing time-explicit methods. This advantage comes, however, with a cost: the solution of a system of nonlinear equations is required at each time step. In this work, the nonlinear system results from the discretization of the hydrodynamical equations with the Crank-Nicholson scheme. We compare the cost of different methods, based on Newton-Raphson iterations, to solve this nonlinear system, and benchmark their performances against time-explicit schemes. Since our general scientific objective is to model stellar interiors, we use as test cases two realistic models for the convective envelope of a red giant and a young Sun. Focusing on 2D simulations, we show that the best performances are obtained with the quasi-Newton method proposed by Broyden. Another important concern is the accuracy of implicit calculations. Based on the study of an idealized problem, namely the advection of a single vortex by a uniform flow, we show that there are two aspects: i) the nonlinear solver has to be accurate enough to resolve the truncation error of the numerical discretization, and ii) the time step has be small enough to resolve the advection of eddies. We show that with these two conditions fulfilled, our implicit methods exhibit similar accuracy to time-explicit schemes, which have lower values for the time step and higher computational costs. Finally, we discuss in the conclusion the applicability of these methods to fully implicit 3D calculations.Comment: Accepted for publication in A&

Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers

In this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by Balsara et al. to construct a new class of computationally efficient high order Lagrangian ADER-WENO one-step ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstruction operator allows the algorithm to achieve high order of accuracy in space, while high order of accuracy in time is obtained by the use of an ADER time-stepping technique based on a local space-time Galerkin predictor. The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the grid, considering the entire Voronoi neighborhood of each node and allows for larger time steps than conventional one-dimensional Riemann solvers. The results produced by the multidimensional Riemann solver are then used twice in our one-step ALE algorithm: first, as a node solver that assigns a unique velocity vector to each vertex, in order to preserve the continuity of the computational mesh; second, as a building block for genuinely multidimensional numerical flux evaluation that allows the scheme to run with larger time steps compared to conventional finite volume schemes that use classical one-dimensional Riemann solvers in normal direction. A rezoning step may be necessary in order to overcome element overlapping or crossing-over. We apply the method presented in this article to two systems of hyperbolic conservation laws, namely the Euler equations of compressible gas dynamics and the equations of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to fourth order of accuracy in space and time have been carried out. Several numerical test problems have been solved to validate the new approach

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

An Asymptotic Preserving Scheme for the ES-BGK model

In this paper, we study a time discrete scheme for the initial value problem of the ES-BGK kinetic equation. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We study an implicit-explicit (IMEX) time discretization in which the convection is explicit while the relaxation term is implicit to overcome the stiffness. We first show how the implicit relaxation can be solved explicitly, and then prove asymptotically that this time discretization drives the density distribution toward the local Maxwellian when the mean free time goes to zero while the numerical time step is held fixed. This naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver for the implicit relaxation term. Moreover, it can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We also show that it is consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. Several numerical examples, in both one and two space dimensions, are used to demonstrate the desired behavior of this scheme

Local stability and a renormalized Newton Method for equilibrium liquid crystal director modeling

We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to problems with a double saddle-point form, for which we have previously proposed a preconditioned nullspace method as an effective solver [A. Ramage and E. C. Gartland, Jr., submitted]. The characterization of local stability of solutions is complicated by the double saddle-point structure, and here we develop efficiently computable criteria in terms of minimum eigenvalues of certain projected Schur complements. We also propose a modified outer iteration (“Renormalized Newton Method”) in which the orientation variables are normalized onto the constraint manifold at each iterative step. This scheme takes advantage of the special structure of these problems, and we prove that it is locally quadratically convergent. The Renormalized Newton Method bears some resemblance to the Truncated Newton Method of computational micromagnetics, and we compare and contrast the two

Analysis of a Darcy-Cahn-Hilliard Diffuse Interface Model for the Hele-Shaw Flow and its Fully Discrete Finite Element Approximation

In this paper we present PDE and finite element analyses for a system of partial differential equations (PDEs) consisting of the Darcy equation and the Cahn-Hilliard equation, which arises as a diffuse interface model for the two phase Hele-Shaw flow. We propose a fully discrete implicit finite element method for approximating the PDE system, which consists of the implicit Euler method combined with a convex splitting energy strategy for the temporal discretization, the standard finite element discretization for the pressure and a split (or mixed) finite element discretization for the fourth order Cahn-Hilliard equation. It is shown that the proposed numerical method satisfies a mass conservation law in addition to a discrete energy law that mimics the basic energy law for the Darcy-Cahn-Hilliard phase field model and holds uniformly in the phase field parameter $\epsilon$. With help of the discrete energy law, we first prove that the fully discrete finite method is unconditionally energy stable and uniquely solvable at each time step. We then show that, using the compactness method, the finite element solution has an accumulation point that is a weak solution of the PDE system. As a result, the convergence result also provides a constructive proof of the existence of global-in-time weak solutions to the Darcy-Cahn-Hilliard phase field model in both two and three dimensions. Finally, we propose a nonlinear multigrid iterative algorithm to solve the finite element equations at each time step. Numerical experiments based on the overall solution method of combining the proposed finite element discretization and the nonlinear multigrid solver are presented to validate the theoretical results and to show the effectiveness of the proposed fully discrete finite element method for approximating the Darcy-Cahn-Hilliard phase field model.Comment: 30 pages, 4 tables, 2 figure