ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step P_NP_M schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate P_NP_M reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Käser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Parés [Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].     

An Analysis of the Dissipation and Dispersion Errors of the P N P M Schemes

We examine the dispersion and dissipation properties of the P _N P _M schemes for linear wave propagation problems. P _N P _M scheme are based on P _N discontinuous Galerkin base approximations augmented with a cell centered polynomial least-squares reconstruction from degree N up to the design polynomial degree M. This methodology can be seen as a generalized discretization framework, as cell centered high order finite volume schemes (N=0) and discontinuous Galerkin schemes (N=M) are included as special cases. We show that with respect to the dispersion error, the pure discontinuous Galerkin variant gives typically the best accuracy for a defined number of points per wavelength. Regarding the dissipation behavior, combinations of N and M exist that result in slightly lower errors for a given resolution. An investigation of the influence of the stencil size on the accuracy of the scheme shows that the errors are smaller the smaller the stencil size for the reconstruction.     

Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for?Conservation Laws on Triangular Meshes

In Zhang and Shu (J. Comput. Phys. 229:3091–3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and Shu (J. Comput. Phys. 229:8918–8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.     

Simulation of a viscous compressible flow past a circular cylinder with high-order discontinuous Galerkin methods

This paper is devoted to the simulation of viscous compressible flows with high-order accurate discontinuous Galerkin methods on bidimensional unstructured meshes. The formulation for the solution of the Navier–Stokes equations is due to Oden et al. [An hp-adaptive discontinuous finite element method for computational fluid dynamics. PhD thesis, The University of Texas at Austin, 1997; J Comput Phys 1998;146:491–519]. It involves a weak imposition of continuity conditions on the state variables and on fluxes across interelement boundaries. It does not make use of any auxiliary variables and then the cost for the implementation is reasonable. The method is coupled with a limiting procedure recently developed by the authors to suppress oscillations near large gradients. The limiter is totally free of problem dependence and maintains the convergence order for errors measured in the L¹-norm. This paper presents detailed numerical results of a viscous compressible flow past a circular cylinder at a Reynolds number of 100 for the cases of subsonic and supersonic regimes. The proposed simulations suggest that the method is very robust and is able to produce very accurate results on unstructured meshes.     

High-speed compressible flow solutions by adaptive cell-centered upwinding algorithm with modified H-correction entropy fix

Adaptive Delaunay triangulation is combined with the cell-centered upwinding algorithm to analyze high-speed compressible flow problems. The H-correction entropy fix is modified and included in the upwinding algorithm for unstructured triangular meshes to improve the computed shock wave resolution. The solution accuracy is further improved by coupling an error estimation procedure to a remeshing algorithm. Efficiency of the combined procedure is evaluated by analyzing supersonic shocks and shock propagation behaviors for both the steady and unsteady high-speed compressible flows.     

Hermite WENO Schemes and Their Application as Limiters for Runge-Kutta Discontinuous Galerkin Method,III: Unstructured Meshes

In [J. Comput. Phys. 193:115–135, 2004] and [Comput. Fluids 34:642–663, 2005], Qiu and Shu developed a class of high order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving nonlinear hyperbolic conservation law systems, and applied them as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods on structured meshes. In this continuation paper, we extend the method to solve two dimensional problems on unstructured meshes. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods. The research was partially supported by the European project ADIGMA on the development of innovative solution algorithms for aerodynamic simulations, NSFC grant 10671091 and JSNSF BK2006511.     

A p-multigrid spectral difference method for two-dimensional unsteady incompressible Navier-Stokes equations

This paper presents the development of a 2D high-order solver with spectral difference method for unsteady incompressible Navier-Stokes equations accelerated by a p-multigrid method. This solver is designed for unstructured quadrilateral elements. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditional characteristics-based schemes. The viscous fluxes are computed using the averaging approach (Sun et al., 2007; Kopriva, 1998) [29] and [12]. A dual time stepping scheme is implemented to deal with physical time marching. A p-multigrid method is implemented (Liang et al., 2009) [16] in conjunction with the dual time stepping method for convergence acceleration. The incompressible SD (ISD) method added with the ACM (SD-ACM) is able to accurately simulate 2D steady and unsteady viscous flows.     

Simulating Compressible Two-Medium Flows with Sharp-Interface Adaptive Runge–Kutta Discontinuous Galerkin Methods

A cut cell based sharp-interface Runge–Kutta discontinuous Galerkin method, with quadtree-like adaptive mesh refinement, is developed for simulating compressible two-medium flows with clear interfaces. In this approach, the free interface is represented by curved cut faces and evolved by solving the level-set equation with high order upstream central scheme. Thus every mixed cell is divided into two cut cells by a cut face. The Runge–Kutta discontinuous Galerkin method is applied to calculate each single-medium flow governed by the Euler equations. A two-medium exact Riemann solver is applied on the cut faces and the Lax–Friedrichs flux is applied on the regular faces. Refining and coarsening of meshes occur according to criteria on distance from the material interface and on magnitudes of pressure/density gradient, and the solutions and fluxes between upper-level and lower-level meshes are synchronized by \(L^2\) projections to keep conservation and high order accuracy. This proposed method inherits the advantages of the discontinuous Galerkin method (compact and high order) and cut cell method (sharp interface and curved cut face), thus it is fully conservative, consistent, and is very accurate on both interface and flow field calculations. Numerical tests with a variety of parameters illustrate the accuracy and robustness of the proposed method.     

Implicit linearity preservation type schemes for moving meshes

We are concerned with the accurate implicit approximation of compressible flows in a fixed and moving mesh context, such as piston engine flows. Geometries are commonly complex and flows compressible. Therefore, it is convenient to develop the numerical approach in the context of a space-time finite-volume formulation for unstructured meshes. The hyperbolic flux is obtained by a generalized Riemann solver taking into account the mesh motion. Using the linearity preservation property we propose a new class of stable implicit schemes developing low numerical viscosity. These schemes can be viewed as a correction of the usual MUSCL flux, induced by the time derivative and mesh motion. Accurate numerical results are obtained for transonic (shock tube) as well as low Mach number flows (diesel engine). It is numerically proved, that for large time steps, those approximations can be as accurate as some explicit schemes. The proposed schemes, due the compactness of the stencils, are well adapted for parallelization strategy.     

A parallel, high-order discontinuous Galerkin code for laminar and turbulent flows

In this study we present a solution method for the compressible Navier-Stokes equations as well as the Reynolds-averaged Navier-Stokes equations (RANS) based on a discontinuous Galerkin (DG) space discretisation. For the turbulent computations we use the standard Wilcox k-ω or the Spalart-Allmaras model in order to close the RANS system. We currently apply either a local discontinuous Galerkin (LDG) or one of the Bassi-Rebay formulations (BR2) for the discretisation of second-order viscous terms. Both approaches (LDG and BR2) can be advanced explicitly as well as implicitly in time by classical integration methods. The boundary is approximated in a continously differentiable fashion by curved elements not to spoil the high-order of accuracy in the interior of the flow field.Computations are performed for the circular cylinder, the flat plate and classical airfoil sections like NACA0012. We compare our obtained results with experimental and computational data as well as analytical (boundary layer) predictions for the flat plate. The excellent parallelisation characteristics of the scheme are demonstrated, achieved by hiding communication latency behind computation.     

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.     

Time-accurate Navier-Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier-Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart-Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade-vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.     

Compact Direct Flux Reconstruction for Conservation Laws

In this study, a high-order discontinuous compact direct flux reconstruction (CDFR) method is developed to solve conservation laws numerically on unstructured meshes. Without explicitly constructing any polynomials, the CDFR method directly calculates the nodal flux derivatives via the compact finite difference approach within elements. To achieve an efficient implementation, the construction procedure of flux derivatives is conducted in a standard element. As a result, a standard flux-derivative-construction matrix can be formulated. The nodal flux derivatives can be directly constructed through the multiplication of this matrix and the flux vectors. It is observed that the CDFR method is identical with the direct flux reconstruction method and the nodal flux reconstruction–discontinuous Galerkin method if Gauss–Legendre points are selected as solution points for degrees p up to 8 tested. A von Neumann analysis is then performed on the CDFR method to demonstrate its linear stability as well as dissipation and dispersion properties for linear wave propagation. Finally, numerical tests are conducted to verify the performance of the CDFR method on solving both steady and unsteady inviscid flows, including those over curved boundaries.     

Discontinuous Galerkin error estimation for hyperbolic problems on unstructured triangular meshes

We extend the error analysis of Adjerid and Baccouch [1], [2] for the discontinuous Galerkin discretization error to variable-coefficient linear hyperbolic problems as well as nonlinear hyperbolic problems on unstructured meshes. We further extend this analysis to transient hyperbolic problems and prove that the local superconvergence results presented in [1] still hold for both steady and transient variable-coefficient linear and nonlinear problems. This local error analysis allows us to construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on each element of general unstructured meshes. We illustrate the superconvergence and the efficiency of our a posteriori error estimates by showing computational results for several linear and nonlinear numerical examples.     

The Mortar-Discontinuous Galerkin Method for the 2D Maxwell Eigenproblem

We consider discontinuous Galerkin (DG) approximations of the Maxwell eigenproblem on meshes with hanging nodes. It is known that while standard DG methods provide spurious-free and accurate approximations on the so-called k-irregular meshes, they may generate spurious solutions on general irregular meshes. In this paper we present a mortar-type method to cure this problem in the two-dimensional case. More precisely, we introduce a projection based penalization at non-conforming interfaces and prove that the obtained DG methods are spectrally correct. The theoretical results are validated in a series of numerical experiments on both convex and non convex problem domains, and with both regular and discontinuous material coefficients.     

Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes

The shallow water equations model flows in rivers and coastal areas and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. In “Xing et al. Adv. Water Resourc. 33: 1476–1493, 2010)”, the authors constructed high order discontinuous Galerkin methods for the shallow water equations which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. In this paper, we explore the extension of these methods on unstructured triangular meshes. The simple positivity-preserving limiter is reformulated, and we prove that the resulting scheme guarantees the positivity of the water depth. Extensive numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.     

Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.     

An hp-Analysis of the Local Discontinuous Galerkin Method for Diffusion Problems

We present an hp-error analysis of the local discontinuous Galerkin method for diffusion problems, considering unstructured meshes with hanging nodes and two- and three-dimensional domains. Our estimates are optimal in the meshsize h and slightly suboptimal in the polynomial approximation order p. Optimality in p is achieved for matching grids and polynomial boundary conditions.     

Application of a posteriori error estimation to finite element simulation of compressible Navier-Stokes flow

In this paper, we discuss how to improve the adaptive finite element simulation of compressible Navier-Stokes flow via a posteriori error estimate analysis. We use the moving space-time finite element method to globally discretize the time-dependent Navier-Stokes equations on a series of adapted meshes. The generalized compressible Stokes problem, which is the Stokes problem in its most generalized form, is presented and discussed. On the basis of the a posteriori error estimator for the generalized compressible Stokes problem, a numerical framework of a posteriori error estimation is established corresponding to the case of compressible Navier-Stokes equations. Guided by the a posteriori errors estimation, a combination of different mesh adaptive schemes involving simultaneous refinement/unrefinement and point-moving are applied to control the finite element mesh quality. Finally, a series of numerical experiments will be performed involving the compressible Stokes and Navier-Stokes flows around different aerodynamic shapes to prove the validity of our mesh adaptive algorithms.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.