Space-time Discontinuous Galerkin Method Research Articles

A space–time discontinuous Galerkin method for linear convection-dominated Sobolev equations

This article presents a space–time discontinuous Galerkin (DG) finite element method for linear convection-dominated Sobolev equations. The finite element method has basis functions that are continuous in space and discontinuous in time, and variable spatial meshes and time steps are allowed. In the discrete intervals of time, using properties of the Radau quadrature rule, eliminates the restriction to space–time meshes of convectional space–time Galerkin methods. The existence and uniqueness of the approximate solution are proved. An optimal priori error estimate in L ∞ ( H 1 ) is derived. Numerical experiments are presented to confirm theoretical results.

A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain

AbstractIn recent years, the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation. This has led to the development of new concepts including the discontinuous enrichment method. Its basic principle is the incorporation of features of the governing partial differential equation in the approximation. In this contribution, this concept is adapted for the simulation of transient problems governed by the wave equation. We present a space–time discontinuous Galerkin method with Lagrange multipliers, where the shape approximation in space and time is based on solutions of the homogeneous wave equation. The use of hierarchical wave‐like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations. Numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space–time finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments.

A spacetime discontinuous Galerkin method for hyperbolic heat conduction

Non-Fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation. For applications involving very short length or time scales, hyperbolic conduction models better represent the physical thermal transport processes. This paper reviews the Maxwell–Cattaneo–Vernotte modification of the Fourier conduction law and describes its implementation within a spacetime discontinuous Galerkin (SDG) finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time. A causal, advancing-front meshing procedure enables a patch-wise solution procedure with linear complexity in the number of spacetime elements. An h-adaptive scheme and a special SDG shock-capturing operator accurately resolve sharp solution features in both space and time. Numerical results for one spatial dimension demonstrate the convergence properties of the SDG method as well as the effectiveness of the shock-capturing method. Simulations in two spatial dimensions demonstrate the proposed method’s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place.

Adaptive spacetime meshing for discontinuous Galerkin methods

Spacetime-discontinuous Galerkin (SDG) finite element methods are used to solve hyperbolic spacetime partial differential equations (PDEs) to accurately model wave propagation phenomena arising in important applications in science and engineering. Tent Pitcher is a specialized algorithm, invented by Üngör and Sheffer (2000) and extended by Erickson et al. (2005) to construct an unstructured simplicial ( d + 1 ) -dimensional spacetime mesh over an arbitrary d-dimensional space domain. Tent Pitcher is an advancing front algorithm that incrementally adds groups of elements to the evolving spacetime mesh. It supports an accurate, local, and parallelizable solution strategy by interleaving mesh generation with an SDG solver. When solving nonlinear PDEs, previous versions of Tent Pitcher must make conservative worst-case assumptions about the physical parameters which limit the duration of spacetime elements. Thus, these algorithms create a mesh with many more elements than necessary. In this paper, we extend Tent Pitcher to give the first spacetime meshing algorithm suitable for efficient simulation of nonlinear phenomena using SDG methods. We adapt the duration of spacetime elements to changing physical parameters due to nonlinear response. Given a triangulated 2-dimensional Euclidean space domain M corresponding to time t = 0 and initial and boundary conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured tetrahedral mesh in the spacetime domain E 2 × R . For every target time T ⩾ 0 , our algorithm meshes the spacetime volume M × [ 0 , T ] with a bounded number of non-degenerate tetrahedra. A recent extension of Tent Pitcher due to Abedi et al. (2004) adapts the spatial size of spacetime elements in 2D × time to a posteriori estimates of numerical error. Our extension of Tent Pitcher retains the ability to perform adaptive refinement and coarsening of the mesh. We thus obtain the first adaptive nonlinear Tent Pitcher algorithm to build spacetime meshes in 2D × time.

Sub‐cell shock capturing and spacetime discontinuity tracking for nonlinear conservation laws

AbstractWe describe two approaches to resolving shocks and other discontinuities in spacetime discontinuous Galerkin (SDG) methods for nonlinear conservation laws. The first is an adaptation of the sub‐cell shock‐capturing technique, recently introduced by Persson and Peraire, to the special circumstances of SDG solutions constructed on causal spacetime grids. We restrict the stabilization operator to spacetime element interiors, thereby preserving the 𝒪(N) computational complexity and the element‐wise conservation properties of the basic SDG method, and use a special discontinuity indicator to limit the stabilization to elements traversed by discontinuous solution features. The method resolves discontinuities within individual spacetime elements having a sufficiently high‐order basis. Numerical studies demonstrate the combination of sub‐cell shock capturing with h‐adaptive spacetime meshing that circumvents the projection errors inherent to purely spatial remeshing procedures.In a second method, we use adaptive spacetime meshing operations to track the trajectories of singular surfaces while maintaining the quality of the surrounding mesh. We present a series of feasibility studies where we track shocks and contact discontinuities in solutions to the inviscid Euler equations, including an example where the trajectories are not known a priori. The SDG‐tracking method sharply resolves discontinuities without mesh refinement and requires very little stabilization. Copyright © 2008 John Wiley & Sons, Ltd.

Space–time discontinuous Galerkin method for dynamics of solids

AbstractIn this study, a space–time discontinuous Galerkin formulation is derived for dynamic analysis of solids. Numerical implementation of proposed formulation is presented, and numerical examples are provided for shock wave propagation problems. Numerical results are compared with that of the known exact solutions of the problems and numerical results obtained from a commercial finite element code. The proposed space–time discontinuous Galerkin formulation is found to be robust and successful in modelling transient problems, i.e. shock wave propagation problems. Copyright © 2007 John Wiley & Sons, Ltd.

A space–time discontinuous Galerkin method for the time-dependent Oseen equations

A space–time discontinuous Galerkin finite element method for the Oseen equations on time-dependent flow domains is presented. The algorithm results in a higher order accurate conservative discretization on moving and deforming meshes and is well suited for hp-adaptation. A detailed analysis of the stability of the numerical discretization is given which shows that the algorithm is unconditionally stable, also when equal order polynomial basis functions for the pressure and velocity are used. The accuracy of the space–time discretization is investigated using a detailed hp-error analysis and computations on a model problem.

Space-Time Discontinuous Galerkin Method for Linear Elastodynamics

Among the time-integration methods used in solid mechanics, Newmark family of algorithms (along with Galerkin finite element) has been most popular and has gathered much attention over the years. Recently, the time discontinuous Galerkin method (TDG) and discontinuous Galerkin method (DGM) are being investigated by the researchers due to their advantages in wave propagation problems. In this study, TDG and DGM are combined to have space-time discontinuous Galerkin formulation for the first time for solid mechanics problems. The basic formulations are presented and some examples are studied.

Space–time discontinuous Galerkin discretization of rotating shallow water equations

A space–time discontinuous Galerkin (DG) discretization is presented for the (rotating) shallow water equations over varying topography. We formulate the space–time DG finite element discretization in an efficient and conservative discretization. The HLLC flux is used as numerical flux through the finite element boundaries. When discontinuities are present, we locally apply dissipation around these discontinuities with the help of Krivodonova’s discontinuity indicator such that spurious oscillations are suppressed. The non-linear algebraic system resulting from the discretization is solved using a pseudo-time integration with a second-order five-stage Runge–Kutta method. A thorough verification of the space–time DG finite element method is undertaken by comparing numerical and exact solutions. We also carry out a discrete Fourier analysis of the one-dimensional linear rotating shallow water equations to show that the method is unconditionally stable with minimal dispersion and dissipation error. The numerical scheme is validated in a novel way by considering various simulations of bore–vortex interactions in combination with a qualitative analysis of PV generation by non-uniform bores. Finally, the space–time DG method is particularly suited for problems where dynamic grid motion is required. To demonstrate this we simulate waves generated by a wave maker and verify these for low amplitude waves where linear theory is approximately valid.

Space–time discontinuous Galerkin method for nonlinear water waves

A space–time discontinuous Galerkin (DG) finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented. The space–time DG method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolution. The algorithm is higher order accurate, both in space and time, and closely related to an arbitrary Lagrangian Eulerian (ALE) approach. A detailed derivation of the numerical algorithm is given including an efficient procedure to solve the nonlinear algebraic equations resulting from the space–time discretization. Numerical examples are shown on a series of model problems to demonstrate the accuracy and capabilities of the method.

Space–time discontinuous Galerkin method for the compressible Navier–Stokes equations

A space–time discontinuous Galerkin finite element method for the compressible Navier–Stokes equations is presented. We explain the space–time setting, derive the weak formulation and discuss our choices for the numerical fluxes. The resulting numerical method allows local grid adaptation as well as moving and deforming boundaries, which we illustrate by computing the flow around a 3D delta wing on an adapted mesh and by simulating the dynamic stall phenomenon of a 2D airfoil in rapid pitch-up maneuver.

Space–time discontinuous Galerkin method for advection–diffusion problems on time-dependent domains

This article presents a space–time discontinuous Galerkin (DG) finite element discretization of the advection–diffusion equation on time-dependent domains. In the space–time DG discretization no distinction is made between the space and time variables and discontinuous basis functions are used both in space and time. This approach results in an efficient numerical technique for physical applications which require moving and deforming elements, is suitable for hp-adaptation and results in a fully conservative discretization. A complete derivation of the space–time DG method for the advection–diffusion equation is given, together with the relation of the space–time discretization with the arbitrary Lagrangian Eulerian (ALE) approach. Detailed proofs of stability and error estimates are also provided. The space–time DG method is demonstrated with numerical experiments that agree well with the error analysis.

An h-adaptive spacetime-discontinuous Galerkin method for linear elastodynamics

We present an h-adaptive version of the spacetime-discontinuous Galerkin (SDG) finite element method for linearized elastodynamics (Abedi et al., 2006). The adaptive version inherits key properties of the basic SDG formulation, including element-wise balance of linear and angular momentum, complexity that is linear in the number of elements and oscillationfree shock capturing. Unstructured spacetime grids allow simultaneous adaptation in space and time. A localized patch-by-patch solution process limits the cost of reanalysis when the error indicator calls for more refinement. Numerical examples demonstrate the method’s performance and shock-capturing capabilities.

A space–time discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance

We present a new space–time discontinuous Galerkin finite element method for linearized elastodynamics that delivers exact balance of linear and angular momentum over every space–time element. The method is formulated for use with fully unstructured space–time grids and uses displacement basis functions that are discontinuous across all inter-element boundaries. We introduce a new space–time formulation of continuum elastodynamics that uses differential forms and the exterior calculus on manifolds to generate a system of space–time field equations and jump conditions. Then we invoke a Bubnov–Galerkin weighted residuals procedure to formulate the finite element method. We describe an implementation on patch-wise causal meshes that features linear complexity in the number of elements and special per-pixel accurate visualization. Numerical examples confirm an a priori error estimate and demonstrate the method’s shock-capturing capabilities.

A Discontinuous Galerkin Method for Linear Symmetric Hyperbolic Systems in Inhomogeneous Media

The Discontinuous Galerkin (DG) method provides a powerful tool for approximating hyperbolic problems. Here we derive a new space-time DG method for linear time dependent hyperbolic problems written as a symmetric system (including the wave equation and Maxwell's equations). The main features of the scheme are that it can handle inhomogeneous media, and can be time-stepped by solving a sequence of small linear systems resulting from applying the method on small collections of space-time elements. We show that the method is stable provided the space-time grid is appropriately constructed (this corresponds to the usual time-step restriction for explicit methods, but applied locally) and give an error analysis of the scheme. We also provide some simple numerical tests of the algorithm applied to the wave equation in two space dimensions (plus time).

A Space-Time Discontinuous Galerkin Method for Convection and Diffusion in Injection Moulding

In this article, we investigate a space-time finite element method for solving the convection-diffusion-reaction equation. This method is based on a discontinuous Galerkin technique using low order elements in space and high order elements in time. The suggested method is applied to solve the transport equation and the heat equation in 3D mold filling. An original mixed approach is investigated to solve the steady diffusion equation. The proposed numerical scheme is validated through analytical examples, then applied to 3D industrial simulations in injection molding process. Several examples show that the computed solutions are stable, robust and fast.

A spacetime discontinuous Galerkin method for scalar conservation laws

We present a spacetime discontinuous Galerkin (SDG) finite element method for scalar hyperbolic conservation laws. The method is consistent with the integral forms of the conservation law and its associated entropy condition written in terms of the physical Godunov flux on each spacetime element. The discrete basis functions are piecewise continuous in spacetime, admitting discontinuities across element boundaries. The resulting Bubnov–Galerkin method is high-order stable for both convex and non-convex flux functions; it does not require stabilization beyond the basic Galerkin projection. The SDG method is applicable to both layered and unstructured spacetime grids. An element-by-element solution scheme delivers O( N) computational complexity (where N is the number of elements) when applied to spacetime meshes that conform to a special causality constraint. We employ two different limiters to control the local overshoot and undershoot that the basic SDG projection generates near shocks.

Convergence of an explicit finite volume scheme for first order symmetric systems

This paper is devoted to the derivation of a O(h1/2) error estimate for the classical upwind, explicit in time, finite volume scheme for linear first order symmetric systems. Such a result already existed for the corresponding implicit in time finite volume scheme, since it can be interpreted as a particular case of the space-time discontinuous Galerkin method but the technique of proof, used in that case, does not extend to explicit schemes. The general framework, recently developed to analyse the convergence rate of finite volume schemes for non linear scalar conservation laws, can not be used either, because it is not adapted for systems, even linear. In this article, we propose a new technique, which takes advantage of the linearity of the problem. The first step consists in controlling the approximation error ∥u−uh∥L2 by an expression of the form −2 , where u is the exact solution, g is a particular smooth function, and μh, νh are some linear forms depending on the approximate solution uh. The second step consists in carefully estimating the error terms and , by using uniform stability results for the discrete problem and regularity properties of the continuous solution.

PITCHING TENTS IN SPACE-TIME: MESH GENERATION FOR DISCONTINUOUS GALERKIN METHOD

Space-time discontinuous Galerkin (DG) methods provide a solution for a wide variety of numerical problems such as inviscid Burgers equation and elastodynamic analysis. Recent research shows that it is possible to solve a DG system using an element-by-element procedure if the space-time mesh satisfies a cone constraint. This constraint requires that the dihedral angle of each interior mesh face with respect to the space domain is less than or equal to a specified angle function α(). Whenever there is a face that violates the cone constraint, the elements at the face must be coupled in the solution. In this paper we consider the problem of generating a simplicial space-time mesh where the size of each group of elements that need to be coupled is bounded by a constant number k. We present an algorithm for generating such meshes which is valid for any nD×TIME domain (n is a natural number). The k in the algorithm is based on a node degree in an n-dimensional space domain mesh.