h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Being implicit in time, the space-time discontinuous Galerkin discretization of the compressible Navier–Stokes equations requires the solution of a non-linear system of algebraic equations at each time-step. The overall performance, therefore, highly depends on the efficiency of the solver. In this article, we solve the system of algebraic equations with a h-multigrid method using explicit Runge–Kutta relaxation. Two-level Fourier analysis of this method for the scalar advection–diffusion equation shows convergence factors between 0.5 and 0.75. This motivates its application to the 3D compressible Navier–Stokes equations where numerical experiments show that the computational effort is significantly reduced, up to a factor 10 w.r.t. single-grid iterations.     

An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations

We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier–Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure–velocity formulation of the incompressible Navier–Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k + 1 in the L²-norm, when polynomials of degree k?0 are used for all components of the approximate solution. And third, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H (div)-conforming, and converges with order k + 2 for k ? 1 and with order 1 for k = 0 in the L²-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.     

A reconstructed discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids

A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier–Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier–Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on arbitrary grids. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG method is able to deliver the same accuracy as the well-known Bassi–Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier–Stokes equations, clearly demonstrating its superior performance over the existing DG methods for solving the compressible Navier–Stokes equations.     

Local discontinuous Galerkin methods for the Cahn–Hilliard type equations

In this paper, we develop local discontinuous Galerkin (LDG) methods for the fourth order nonlinear Cahn–Hilliard equation and system. The energy stability of the LDG methods is proved for the general nonlinear case. Numerical examples for the Cahn–Hilliard equation and the Cahn–Hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods.     

The Elastoplast Discontinuous Galerkin (EDG) method for the Navier–Stokes equations

The present work details the Elastoplast (this name is a translation from the French “sparadrap”, a concept first applied by Yves Morchoisne for Spectral methods [1]) Discontinuous Galerkin (EDG) method to solve the compressible Navier–Stokes equations. This method was first presented in 2009 at the ICOSAHOM congress with some Cartesian grid applications. We focus here on unstructured grid applications for which the EDG method seems very attractive. As in the Recovery method presented by van Leer and Nomura in 2005 for diffusion, jumps across element boundaries are locally eliminated by recovering the solution on an overlapping cell. In the case of Recovery, this cell is the union of the two neighboring cells and the Galerkin basis is twice as large as the basis used for one element. In our proposed method the solution is rebuilt through an L² projection of the discontinuous interface solution on a small rectangular overlapping interface element, named Elastoplast, with an orthogonal basis of the same order as the one in the neighboring cells. Comparisons on 1D and 2D scalar diffusion problems in terms of accuracy and stability with other viscous DG schemes are first given. Then, 2D results on acoustic problems, vortex problems and boundary layer problems both on Cartesian or unstructured triangular grids illustrate stability, precision and versatility of this method.     

On physics-based preconditioning of the Navier–Stokes equations

We develop a fully implicit scheme for the Navier–Stokes equations, in conservative form, for low to intermediate Mach number flows. Simulations in this range of flow regime produce stiff wave systems in which slow dynamical (advective) modes coexist with fast acoustic modes. Viscous and thermal diffusion effects in refined boundary layers can also produce stiffness. Implicit schemes allow one to step over the fast wave phenomena (or unresolved viscous time scales), while resolving advective time scales. In this study we employ the Jacobian-free Newton–Krylov (JFNK) method and develop a new physics-based preconditioner. To aid in overcoming numerical stiffness caused by the disparity between acoustic and advective modes, the governing equations are transformed into the primitive-variable form in a preconditioning step. The physics-based preconditioning incorporates traditional semi-implicit and physics-based splitting approaches without a loss of consistency between the original and preconditioned systems. The resulting algorithm is capable of solving low-speed natural circulation problems (M∼10^-4)$(M\sim {10}^{-4})$ with significant heat flux as well as intermediate speed (M∼1)$(M\sim 1)$ flows efficiently by following dynamical (advective) time scales of the problem.     

Numerical comparisons of time–space iterative method and spatial iterative methods for the stationary Navier–Stokes equations

This paper makes some numerical comparisons of time–space iterative method and spatial iterative methods for solving the stationary Navier–Stokes equations. The time–space iterative method consists in solving the nonstationary Stokes equations based on the time–space discretization by the Euler implicit/explicit scheme under a weak uniqueness condition (A2). The spatial iterative methods consist in solving the stationary Stokes scheme, Newton scheme, Oseen scheme based on the spatial discretization under some strong uniqueness assumptions. We compare the stability and convergence conditions of the time–space iterative method and the spatial iterative methods. Moreover, the numerical tests show that the time–space iterative method is the more simple than the spatial iterative methods for solving the stationary Navier–Stokes problem. Furthermore, the time–space iterative method can solve the stationary Navier–Stokes equations with some small viscosity and the spatial iterative methods can only solve the stationary Navier–Stokes equations with some large viscosities.     

Accuracy analysis of explicit Runge–Kutta methods applied to the incompressible Navier–Stokes equations

This paper investigates the temporal accuracy of the velocity and pressure when explicit Runge–Kutta methods are applied to the incompressible Navier–Stokes equations. It is shown that, at least up to and including fourth order, the velocity attains the classical order of accuracy without further constraints. However, in case of a time-dependent gradient operator, which can appear in case of time-varying meshes, additional order conditions need to be satisfied to ensure the correct order of accuracy. Furthermore, the pressure is only first-order accurate unless additional order conditions are satisfied. Two new methods that lead to a second-order accurate pressure are proposed, which are applicable to a certain class of three- and four-stage methods. A special case appears when the boundary conditions for the continuity equation are independent of time, since in that case the pressure can be computed to the same accuracy as the velocity field, without additional cost. Relevant computations of decaying vortices and of an actuator disk in a time-dependent inflow support the analysis and the proposed methods.     

Spectrally-accurate algorithm for moving boundary problems for the Navier–Stokes equations

A spectral algorithm based on the immersed boundary conditions (IBC) concept is developed for simulations of viscous flows with moving boundaries. The algorithm uses a fixed computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization uses Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. It has been demonstrated that the algorithm delivers the theoretically predicted accuracy in both time and space. Performances of various linear solvers employed in the solution process have been evaluated and a new class of solver that takes advantage of the structure of the coefficient matrix has been proposed. The new solver results in a significant acceleration of computations as well as in a substantial reduction in memory requirements.     

Stabilization and scalable block preconditioning for the Navier–Stokes equations

This study compares several block-oriented preconditioners for the stabilized finite element discretization of the incompressible Navier–Stokes equations. This includes standard additive Schwarz domain decomposition methods, aggressive coarsening multigrid, and three preconditioners based on an approximate block LU factorization, specifically SIMPLEC, LSC, and PCD. Robustness is considered with a particular focus on the impact that different stabilization methods have on preconditioner performance. Additionally, parallel scaling studies are undertaken. The numerical results indicate that aggressive coarsening multigrid, LSC and PCD all have good algorithmic scalability. Coupling this with the fact that block methods can be applied to systems arising from stable mixed discretizations implies that these techniques are a promising direction for developing scalable methods for Navier–Stokes.     

Third order residual distribution schemes for the Navier–Stokes equations

We construct a third order multidimensional upwind residual distribution scheme for the system of the Navier–Stokes equations. The underlying approximation is obtained using standard P² Lagrange finite elements. To discretise the inviscid component of the equations, each element is divided in sub-elements over which we compute a high order residual defined as the integral of the inviscid fluxes on the boundary of the sub-element. The residuals are distributed to the nodes of each sub-element in a multi-dimensional upwind way. To obtain a discretisation of the viscous terms consistent with this multi-dimensional upwind approach, we make use of a Petrov–Galerkin analogy. The analogy allows to find a family of test functions which can be used to obtain a weak approximation of the viscous terms. The performance of this high-order method is tested on flows with high and low Reynolds number.     

Analogue of the Cole-Hopf transform for the incompressible Navier–Stokes equations and its application

We consider the Navier–Stokes equations written in the stream function in two dimensions and vector potentials in three dimensions, which are critical dependent variables. On this basis, we introduce an analogue of the Cole-Hopf transform, which exactly reduces the Navier–Stokes equations to the heat equations with a potential term (i.e. the nonlinear Schrödinger equation at imaginary times). The following results are obtained. (i) A regularity criterion immediately obtains as the boundedness of condition for the potential term when the equations are recast in a path-integral form by the Feynman-Kac formula. (ii) This in turn gives an additional characterisation of possible singularities for the Navier–Stokes equations. (iii) Some numerical results for the two-dimensional Navier–Stokes equations are presented to demonstrate how the potential term captures near-singular structures. Finally, we extend this formulation to higher dimensions, where the regularity issues are markedly open.     

Compact fourth-order finite volume method for numerical solutions of Navier–Stokes equations on staggered grids

The development of a compact fourth-order finite volume method for solutions of the Navier–Stokes equations on staggered grids is presented. A special attention is given to the conservation laws on momentum control volumes. A higher-order divergence-free interpolation for convective velocities is developed which ensures a perfect conservation of mass and momentum on momentum control volumes. Three forms of the nonlinear correction for staggered grids are proposed and studied. The accuracy of each approximation is assessed comparatively in Fourier space. The importance of higher-order approximations of pressure is discussed and numerically demonstrated. Fourth-order accuracy of the complete scheme is illustrated by the doubly-periodic shear layer and the instability of plane-channel flow. The efficiency of the scheme is demonstrated by a grid dependency study of turbulent channel flows by means of direct numerical simulations. The proposed scheme is highly accurate and efficient. At the same level of accuracy, the fourth-order scheme can be ten times faster than the second-order counterpart. This gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost.     

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations

The Vlasov–Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. In particular, we develop in this work a method that discretizes the 1 + 1 Vlasov–Poisson system via a high-order discontinuous Galerkin (DG) method in phase space, and an operator split, semi-Lagrangian method in time. Second-order accuracy in time is relatively easy to achieve via Strang operator splitting. With additional work, using higher-order splitting and a higher-order method of characteristics, we also demonstrate how to push this scheme to fourth-order accuracy in time. We show how to resolve all of the Lagrangian dynamics in such a way that mass is exactly conserved, positivity is maintained, and high-order accuracy is achieved. The Poisson equation is solved to high-order via the smallest stencil local discontinuous Galerkin (LDG) approach. We test the proposed scheme on several standard test cases.     

Proper general decomposition (PGD) for the resolution of Navier–Stokes equations

In this work, the PGD method will be considered for solving some problems of fluid mechanics by looking for the solution as a sum of tensor product functions. In the first stage, the equations of Stokes and Burgers will be solved. Then, we will solve the Navier–Stokes problem in the case of the lid-driven cavity for different Reynolds numbers (Re = 100, 1000 and 10,000). Finally, the PGD method will be compared to the standard resolution technique, both in terms of CPU time and accuracy.     

Output-based space–time mesh adaptation for the compressible Navier–Stokes equations

This paper presents an output-based adaptive algorithm for unsteady simulations of convection-dominated flows. A space–time discontinuous Galerkin discretization is used in which the spatial meshes remain static in both position and resolution, and in which all elements advance by the same time step. Error estimates are computed using an adjoint-weighted residual, where the discrete adjoint is computed on a finer space obtained by order enrichment of the primal space. An iterative method based on an approximate factorization is used to solve both the forward and adjoint problems. The output error estimate drives a fixed-growth adaptive strategy that employs hanging-node refinement in the spatial domain and slab bisection in the temporal domain. Detection of space–time anisotropy in the localization of the output error is found to be important for efficiency of the adaptive algorithm, and two anisotropy measures are presented: one based on inter-element solution jumps, and one based on projection of the adjoint. Adaptive results are shown for several two-dimensional convection-dominated flows, including the compressible Navier–Stokes equations. For sufficiently-low accuracy levels, output-based adaptation is shown to be advantageous in terms of degrees of freedom when compared to uniform refinement and to adaptive indicators based on approximation error and the unweighted residual. Time integral quantities are used for the outputs of interest, but entire time histories of the integrands are also compared and found to converge rapidly under the proposed scheme. In addition, the final output-adapted space–time meshes are shown to be relatively insensitive to the starting mesh.     

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo [7]. Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity.     

An Asymptotic-Preserving all-speed scheme for the Euler and Navier–Stokes equations

We present an Asymptotic-Preserving ‘all-speed’ scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretization, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime. The divergence-free condition on the velocity in the incompressible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and two-dimensional numerical results provide a validation of the Asymptotic-Preserving ‘all-speed’ properties.     

Beyond the Navier–Stokes equations: Burnett hydrodynamics

This work is mainly concerned with the extension of hydrodynamics beyond the Navier–Stokes equations, a regime known as Burnett hydrodynamics. The derivation of the Burnett equations is considered from several theoretical approaches. In particular we discuss the Chapman–Enskog, Grad’s method, and Truesdell’s approach for solving the Boltzmann equation. Also, their derivation using the macroscopic approach given by extended thermodynamics is mentioned. The problems and successes of these equations are discussed and some alternatives proposed to improve them are mentioned. Comparisons of the predictions coming from the Burnett equations with experiments and/or simulations are given in order to have the necessary elements to give a critical assessment of their validity and usefulness.