This work extends the flux-corrected transport (FCT) methodology to arbitrary order continuous finite element discretizations of scalar conservation laws on simplex meshes. Using Bernstein ...polynomials as local basis functions, we constrain the total variation of the numerical solution by imposing local discrete maximum principles on the Bézier net. The design of accuracy-preserving FCT schemes for high order Bernstein–Bézier finite elements requires the development of new algorithms and/or generalization of limiting techniques tailored for linear and multilinear Lagrange elements. In this paper, we propose (i) a new discrete upwinding strategy leading to local extremum bounded low order approximations with compact stencils, (ii) high order variational stabilization based on the difference between two gradient approximations, and (iii) new localized limiting techniques for antidiffusive element contributions. The optional use of a smoothness indicator, based on a second derivative test, makes it possible to potentially avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1D and 2D.
The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in ...space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank–Nicolson scheme and backward Euler scheme are utilized.
•Continuous finite element schemes for hyperbolic systems.•Nodal variational limiting scheme for nonlinear scalars and hyperbolic system.•High order background dissipation for phase errors.•Implicit and explicit time integrators.
We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and ...high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly h-coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level solver/preconditioner. Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends on the smoothness of the solution and can offer significant speedups and memory savings compared to the nested dissection direct solver. While the proposed algorithms are developed within the HDG framework, they are applicable to other hybrid(ized) high-order finite element methods. Moreover, we show that our multilevel algorithms can be interpreted as a multigrid method with specific intergrid transfer and smoothing operators. With several numerical examples from Poisson, pure transport, and convection-diffusion equations we demonstrate the robustness and scalability of the algorithms with respect to solution order. While scalability with mesh size in general is not guaranteed and depends on the smoothness of the solution and the type of equation, improving it is a part of future work.
The magnetohydrodynamics (MHD) equations are continuum models used in the study of a wide range of plasma physics systems, including the evolution of complex plasma dynamics in tokamak disruptions. ...However, efficient numerical solution methods for MHD are extremely challenging due to disparate time and length scales, strong hyperbolic phenomena, and nonlinearity. Therefore the development of scalable, implicit MHD algorithms and high-resolution adaptive mesh refinement strategies is of considerable importance. In this work, we develop a high-order stabilized finite-element algorithm for the reduced visco-resistive MHD equations based on the MFEM finite element library (mfem.org). The scheme is fully implicit, solved with the Jacobian-free Newton-Krylov (JFNK) method with a physics-based preconditioning strategy. Our preconditioning strategy is a generalization of the physics-based preconditioning methods in Chacón et al. (2002) 3 to adaptive, stabilized finite elements. Algebraic multigrid methods are used to invert sub-block operators to achieve scalability. A parallel adaptive mesh refinement scheme with dynamic load-balancing is implemented to efficiently resolve the multi-scale spatial features of the system. Our implementation uses the MFEM framework, which provides arbitrary-order polynomials and flexible adaptive conforming and non-conforming meshes capabilities. Results demonstrate the accuracy, efficiency, and scalability of the implicit scheme in the presence of large scale disparity. The potential of the AMR approach is demonstrated on an island coalescence problem in the high Lundquist-number regime (≥107) with the successful resolution of plasmoid instabilities and thin current sheets.
•Fully implicit SUPG-based stabilized finite elements for resistive MHD.•Scalable physics-based preconditioning.•A parallel AMR algorithm with dynamic load balancing in MFEM.•Demonstration of excellent parallel and algorithmic scaling up to 4096 CPUs.•Multi-scale simulations for plasmoid instabilities of high Lundquist number.
•Matrix-free methods for transport and remap problems based on residual distribution.•High-order Bernstein finite element discretizations.•Theoretical foundation that guarantees reasonable, CFL-like ...time step restrictions.•Comparison of new monolithic limiters to Flux-Corrected-Transport.•Optimal convergence rates for smooth solutions achieved by nodal smoothness indicators.
This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. The convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems.
This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in ...the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. The importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. In the case of transient problem, we also observe a reduction in the computational cost.
•A nonlinear stabilization technique for the finite element discretization of Euler equations is proposed.•The method is differentiable, linearly and local bounds preserving for Euler equations.•A continuation method using parameters of the stabilization is also proposed to enhance nonlinear convergence.•Numerical results show an improvement of the robustness and the nonlinear convergence.
Strong stability preserving (SSP) time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain ...time-step restriction. The search for high order strong stability preserving time-stepping methods with high order and large allowable time-step has been an active area of research. It is known that implicit SSP Runge–Kutta methods exist only up to sixth order; however, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge–Kutta methods of any
linear order
. In the current work we find implicit SSP Runge–Kutta methods with high linear order
p
l
i
n
≤
9
and nonlinear orders
p
=
2
,
3
,
4
, that are optimal in terms of allowable SSP time-step. Next, we formulate a novel optimization problem for implicit–explicit (IMEX) SSP Runge–Kutta methods and find optimized IMEX SSP Runge–Kutta pairs that have high linear order
p
l
i
n
≤
7
and nonlinear orders up to
p
=
4
. We also find implicit methods with large linear stability regions that pair with known explicit SSP Runge–Kutta methods. These methods are then tested on sample problems to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems.
This study considers an implicit finite element formulation for an ideal fully-ionized multifluid electromagnetic plasma system. The formulation is based on fully-implicit Runge-Kutta time ...discretizations and a monolithic discrete algebraic flux corrected (AFC) continuous Galerkin (CG) spatial discretization of the coupled system. The AFC approach adds scalar artificial diffusion to the high-order, semi-discrete Galerkin method and uses mass lumping in the time derivative term. The result is a low-order method that attempts to enforce local-extremum-diminishing properties for the hyperbolic system. An element-based iterative limiter is applied to reduce the amount of artificial diffusion that is used in regions where the solution is smooth and the additional stabilization is not required. Two models are considered for the electromagnetics portion of the system: an electrostatic model, and a full Maxwell system with a parabolic divergence cleaning approach that enforces the required involutions on the electric and magnetic fields. Results are presented that demonstrate the accuracy and robustness of the formulation for smooth and discontinuous solutions to challenging plasma physics problems. This includes a demonstration that the solution of the full multifluid system yields the expected behavior in the ideal shock-MHD limit.
•A fully nodal continuous Galerkin discretization of the multifluid plasma system is presented.•The discretization is stabilized using an algebraic flux correction (AFC) method.•A parabolic divergence cleaning approach is used to enforce Gauss' laws.•The full multifluid system is shown to produce the expected behavior in the ideal shock-MHD limit.
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