A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions

In part I of these two papers we introduced for inviscid flow in one space dimension a discontinuous Galerkin scheme of arbitrary order of accuracy in space and time. In the second part we extend the scheme to the compressible Navier-Stokes equations in multi dimensions. It is based on a space-time Taylor expansion at the old time level in which all time or mixed space-time derivatives are replaced by space derivatives using the Cauchy-Kovalevskaya procedure. The surface and volume integrals in the variational formulation are approximated by Gaussian quadrature with the values of the space-time approximate solution. The numerical fluxes at grid cell interfaces are based on the approximate solution of generalized Riemann problems for both, the inviscid and viscous part. The presented scheme has to satisfy a stability restriction similar to all other explicit DG schemes which becomes more restrictive for higher orders. The loss of efficiency, especially in the case of strongly varying sizes of grid cells is circumvented by use of different time steps in different grid cells. The presented time accurate numerical simulations run with local time steps adopted to the local stability restriction in each grid cell. In numerical simulations for the two-dimensional compressible Navier-Stokes equations we show the efficiency and the optimal order of convergence being p+1, when a polynomial approximation of degree p is used.     

Moment preserving schemes for Euler equations

A high order accurate finite difference scheme is proposed for one-dimensional Euler equations. In the scheme a set of first three moments of each signal are preserved during the updating. The scheme is one of 5th order in space and 4th order in time. This feature is different from that in typical existing methods in which the use of the first three polynomials results in only 3rd order accuracy in space. The scheme has different features from the existing high order schemes, and the most noticeable are the simultaneous discretization both in space and time, and the use of moments of Riemann invariants instead of primitive physical variables. Numerical examples are given to show the accuracy of the scheme and its robustness for the flows involving shocks.     

Discontinuous upwind residual distribution: A route to unconditional positivity and high order accuracy

This paper proposes an approach to the approximation of time-dependent hyperbolic conservation laws which is both second order accurate in space and time (for any sufficiently smooth solution profile, even one containing turning points) and free of spurious oscillations for any time-step. The numerical algorithm is based on the concept of fluctuation distribution, applied on a space-time mesh of triangular prisms, for which second order accurate schemes already exist which are oscillation-free if the time-step satisfies a CFL-type constraint. This restriction is lifted here by combining the concept of a two-layer scheme with a representation of the solution which is allowed to be discontinuous-in-time. Numerical results are presented in two space dimensions, using unstructured meshes of space-time triangular prisms, for the scalar advection equation, Burgers’ equation and the Euler equations of gas dynamics.     

A Discontinuous Galerkin Scheme Based on a Space–Time Expansion. I. Inviscid Compressible Flow in One Space Dimension

In this paper, we propose an explicit discontinuous Galerkin scheme for conservation laws which is of arbitrary order of accuracy in space and time. The basic idea is to use a Taylor expansion in space and time to define a space–time polynomial in each space–time element. The space derivatives are given by the approximate solution at the old time level, the time derivatives and the mixed space–time derivatives are computed from these space derivatives using the so-called Cauchy–Kovalevskaya procedure. The space–time volume integral is approximated by Gauss quadrature with values at the space–time Gaussian points obtained from the Taylor expansion. The flux in the surface integral is approximated by a numerical flux with arguments given by the Taylor expansions from the left and from the right-hand side of the element interface. The locality of the presented method together with the space–time expansion gives the attractive feature that the time steps may be different in each grid cell. Hence, we drop the common global time levels and propose that every grid zone runs with its own time step which is determined by the local stability restriction. In spite of the local time steps the scheme is locally conservative, fully explicit, and arbitrary order accurate in space and time for transient calculations. Numerical results are shown for the one-dimensional Euler equations with orders of accuracy one up to six in space and time.     

A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker–Planck equations in space dimensions \(d\ge 2\) is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies and given external potentials, e.g. the porous medium equation and the fast diffusion equation. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. Consistency is analyzed in the planar situation, \(d=2\). A variety of numerical experiments for the porous medium equation indicates that the scheme is well-adapted to track the growth of the solution’s support.     

All Mach Number Second Order Semi-implicit Scheme for the Euler Equations of Gas Dynamics

This paper presents an asymptotic preserving (AP) all Mach number finite volume shock capturing method for the numerical solution of compressible Euler equations of gas dynamics. Both isentropic and full Euler equations are considered. The equations are discretized on a staggered grid. This simplifies flux computation and guarantees a natural central discretization in the low Mach limit, thus dramatically reducing the excessive numerical diffusion of upwind discretizations. Furthermore, second order accuracy in space is automatically guaranteed. For the time discretization we adopt an Semi-IMplicit/EXplicit (S-IMEX) discretization getting an elliptic equation for the pressure in the isentropic case and for the energy in the full Euler case. Such equations can be solved linearly so that we do not need any iterative solver thus reducing computational cost. Second order in time is obtained by a suitable S-IMEX strategy taken from Boscarino et al. (J Sci Comput 68:975–1001, 2016). Moreover, the CFL stability condition is independent of the Mach number and depends essentially on the fluid velocity. Numerical tests are displayed in one and two dimensions to demonstrate performance of our scheme in both compressible and incompressible regimes.     

An Adaptive Time Step Procedure for a Parabolic Problem with Blow-up

In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time. We use standard linear elements with mass lumping for the space variable. For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then, we use explicit Runge-Kutta methods for this equivalent problem. In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case. We obtain necessary and sufficient conditions for the blow-up of the numerical solution and prove that the numerical blow-up time converges to the continuous one. We also study, for the explicit Euler approximation, the localization of blow-up points for the numerical scheme. Received October 4, 2001; revised March 27, 2002 Published online: July 8, 2002     

An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation

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.     

Asymptotic properties of the space–time adaptive numerical solution of a nonlinear heat equation

We consider the fully adaptive space–time discretization of a class of nonlinear heat equations by Rothe’s method. Space discretization is based on adaptive polynomial collocation which relies on equidistribution of the defect of the numerical solution, and the time propagation is realized by an adaptive backward Euler scheme. From the known scaling laws, we infer theoretically the optimal grids implying error equidistribution, and verify that our adaptive procedure closely approaches these optimal grids.     

Numerical solution of a phase field model for polycrystallization processes in binary mixtures

We consider the numerical solution of a phase field model for polycrystallization in the solidification of binary mixtures in a domain \( \varOmega \subset \mathbb {R}^2\). The model is based on a free energy in terms of three order parameters: the local orientation \(\varTheta \) of the crystals, the local crystallinity \(\phi \), and the concentration c of one of the components of the binary mixture. The equations of motion are given by an initial-boundary value problem for a coupled system of partial differential equations consisting of a regularized second order total variation flow in \( \varTheta \), an \(L^2\) gradient flow in \(\phi \), and a \(W^{1,2}(\varOmega )^*\) gradient flow in c. Based on an implicit discretization in time by the backward Euler scheme, we suggest a splitting method such that the three semidiscretized equations can be solved separately and prove existence of a solution. As far as the discretization in space is concerned, the fourth order Cahn–Hilliard type equation in c is taken care of by a \(\hbox {C}^0\) Interior Penalty Discontinuous Galerkin approximation which has the advantage that the same finite element space can be used as well for the spatial discretization of the equations in \( \varTheta \) and \( \phi \). The fully discretized equations represent parameter dependent nonlinear algebraic systems with the discrete time as a parameter. They are solved by a predictor corrector continuation strategy featuring an adaptive choice of the time-step. Numerical results illustrate the performance of the suggested numerical method.

     

Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

This paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems.     

Fully discrete finite element method based on pressure stabilization for the transient Stokes equations

In this work, a new fully discrete stabilized finite element method is studied for the two-dimensional transient Stokes equations. This method is to use the difference between a consistent mass matrix and underintegrated mass matrix as the complement for the pressure. The spatial discretization is based on the P₁–P₁ triangular element for the approximation of the velocity and pressure, the time discretization is based on the Euler semi-implicit scheme. Some error estimates for the numerical solutions of fully discrete stabilized finite element method are derived. Finally, we provide some numerical experiments, compared with other methods, we can see that this novel stabilized method has better stability and accuracy results for the unsteady Stokes problem.     

A new fourth-order difference scheme for solving an N-carrier system with Neumann boundary conditions

In this paper, a numerical method is developed to solve an N-carrier system with Neumann boundary conditions. First, we apply the compact finite difference scheme of fourth order for discretizing spatial derivatives at the interior points. Then, we develop a new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy. Lastly, by using a Padé approximation method for the resulting linear system of ordinary differential equations, a new compact finite difference scheme is obtained. The present scheme has second-order accuracy in time direction and fourth-order accuracy in space direction. It is shown that the scheme is unconditionally stable. The present scheme is tested by two numerical examples, which show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate when compared with those obtained by using other previous methods.     

Meshless local numerical procedure based on interpolating moving least squares approximation and exponential time differencing fourth-order Runge–Kutta (ETDRK4) for solving stochastic parabolic interface problems

The main propose of this investigation is to develop an interpolating meshless numerical procedure for solving the stochastic parabolic interface problems. The present numerical algorithm is constructed from the interpolating moving least squares (ISMLS) approximation. At first, the space variable has been discretized by using the ISMLS approximation. Then, the PDE reduces to the system of nonlinear ODEs. In the next, to achieve a high-order numerical formula, we employ a fourth-order time discrete scheme that is well-known as the explicit fourth-order exponential time differencing Runge-Kutta method (ETDRK4). This method is simple and has acceptable accuracy for solving the considered problems. Several examples with adequate intricacy are examined to check the new numerical procedure.

     

Arbitrary High-Order Discontinuous Galerkin Schemes for the Magnetohydrodynamic Equations

In this paper, we propose a discontinuous Galerkin scheme with arbitrary order of accuracy in space and time for the magnetohydrodynamic equations. It is based on the Arbitrary order using DERivatives (ADER) methodology: the high order time approximation is obtained by a Taylor expansion in time. In this expansion all the time derivatives are replaced by space derivatives via the Cauchy-Kovalevskaya procedure. We propose an efficient algorithm of the Cauchy-Kovalevskaya procedure in the case of the three-dimensional magneto-hydrodynamic (MHD) equations. Parallel to the time derivatives of the conservative variables the time derivatives of the fluxes are calculated. This enables the analytic time integration of the volume integral as well as that of the surface integral of the fluxes through the grid cell interfaces which occur in the discrete equations. At the cell interfaces the fluxes and all their derivatives may jump. Following the finite volume ADER approach the break up of all these jumps into the different waves are taken into account to get proper values of the fluxes at the grid cell interfaces. The approach under considerations is directly based on the expansion of the flux in time in which the leading order term may be any numerical flux calculation for the MHD-equation. Numerical convergence results for these equations up to 7th order of accuracy in space and time are shown.     

An efficient and accurate alternative to subincrementation for elastic-plastic analysis by the finite element method

In this paper a new and efficient alternative to subincrementation is developed for analysis of solid media with rate independent elastic-plastic material behavior. This alternative method is not unlike the subincrementation procedure in that it represents an Euler integration of the nonlinear constitutive equations. However, it takes advantage of the fact that the Euler integration procedure assumes proportional loading steps so that when the uniaxial stress-strain curve is idealized as a piecewise linear relation very large forward integration steps give accurate results. The new procedure, which we call the ζ method, is equally appropriate for cyclic loading with combined isotropic and kinematic hardening. However, due to the nonuniqueness of the monotonic uniaxial stress-strain relation in rate dependent media, the method is not appropriate for use in viscoplastic media. Although the algorithm deals only with the evaluation of a classical plasticity based constitutive law, numerical results are reported herein for an assortment of problems by the finite element method. It is shown via these results that the ζ method discussed herein provides not only accuracy which is superior to the subincrementation method, but the resulting algorithm also shows improved numerical efficiency.     

High Order Multi-dimensional Characteristics Tracing for the Incompressible Euler Equation and the Guiding-Center Vlasov Equation

In this paper, we propose a high order characteristics tracing scheme for the two-dimensional nonlinear incompressible Euler system in vorticity stream function formulation and the guiding center Vlasov model. Such a scheme is incorporated into a semi-Lagrangian finite difference WENO framework for simulating the aforementioned model equations. This is an extension of our earlier work on high order characteristics tracing scheme for the 1D nonlinear Vlasov–Poisson system (Qiu and Russo in J Sci Comput 71:414–434, 2017). The effectiveness of the proposed scheme is demonstrated numerically by an extensive set of test cases.     

Primal Discontinuous Galerkin Methods for Time-Dependent Coupled Surface and Subsurface Flow

This paper introduces and analyzes a numerical method based on discontinuous finite element methods for solving the two-dimensional coupled problem of time-dependent incompressible Navier-Stokes equations with the Darcy equations through Beaver-Joseph-Saffman’s condition on the interface. The proposed method employs Crank-Nicolson discretization in time (which requires one step of a first order scheme namely backward Euler) and primal DG method in space. With the correct assumption on the first time step optimal error estimates are obtained that are high order in space and second order in time.     

Strong Convergence Analysis of the Stochastic Exponential Rosenbrock Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

In this paper, we consider the numerical approximation of a general second order semilinear stochastic spartial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise with finite trace and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square \(L^2\) norm. Numerical experiments to sustain theoretical results are provided.