Space-time Discontinuous Galerkin Research Articles

Enriched space–time finite element method: a new paradigm for multiscaling from elastodynamics to molecular dynamics

SUMMARYThe main objective of this paper is to present an enriched version of the space–time FEM method to incorporate multiple temporal scale features with a focus on dynamics problems. The method is established by integrating the basic framework of the space–time discontinuous Galerkin method with the extended finite element method. Two versions of the method have been developed: one at the continuum scale (elastodynamics) and the other focuses on the dynamics at the molecular level. After an initial outline of the formulation, we explore the incorporation of different types of enrichments based on the length scale of interest. The effects of both continuous and discontinuous enrichments are demonstrated through numerical examples involving wave propagations and dynamic fracture in harmonic lattice. The robustness of the method is evaluated in terms of convergence and the ability to capture the fine scale features. It is shown that the enriched space–time FEM leads to an improvement in the convergence properties over the traditional space–time FEM for problems with multiple temporal features. It is also highly effective in integrating atomistic with continuum representations with a coupled framework. Copyright © 2012 John Wiley & Sons, Ltd.

Extension Of The High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme To Solve Time Dependent Diffusion Equations

AbstractIn this paper, the high-order space-time discontinuous Galerkin cell vertex scheme (DG-CVS) developed by the authors for hyperbolic conservation laws is extended for time dependent diffusion equations. In the extension, the treatment of the diffusive flux is exactly the same as that for the advective flux. Thanks to the Riemann-solver-free and reconstruction-free features of DG-CVS, both the advective flux and the diffusive flux are evaluated using continuous information across the cell interface. As a result, the resulting formulation with diffusive fluxes present is still consistent and does not need any extra ad hoc techniques to cure the common “variational crime” problem when traditional DG methods are applied to diffusion problems. For this reason, DG-CVS is conceptually simpler than other existing DG-typed methods.

Theory of the Space-Time Discontinuous Galerkin Method for Nonstationary Parabolic Problems with Nonlinear Convection and Diffusion

The paper presents the theory of the space-time discontinuous Galerkin finite element method for the discretization of a nonstationary convection-diffusion initial-boundary value problem with nonlinear convection and nonlinear diffusion. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time discretization. In the space discretization the nonsymmetric, symmetric, and incomplete interior and boundary penalty approximations of diffusion terms are used. The paper is concerned with the analysis of error estimates in the “$L^2(L^2)$”- and “DG”-norm formed by the “$L^2(H^1) $”-seminorm and penalty terms. An important ingredient used in the derivation of the error estimate is the concept of the discrete characteristic function and its properties. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step if the Dirichlet boundary condition behaves in time as a polynomial of degree $\leq q$. In a general case, the derived estimates become suboptimal in time.

A splitting mixed space-time discontinuous Galerkin method for parabolic problems

A splitting mixed space-time discontinuous Galerkin method is formulated to solve a class of parabolic problems. This method, in which the stress equation is separated from displacement equation, is based on mixed method and space-time discontinuous finite element method which is discontinuous in time and continuous in space. By a splitting technique, the stress equation is separated from the stress-displacement coupled system. The finite element approximation of the stress is solved by time discontinuous Galerkin method with high accuracy. Then, if required, the discrete displacement function is also solved by the time discontinuous Galerkin method. The convergence of the scheme is analyzed by the technique of combining finite difference and finite element methods. The optimal priori error estimates in norm for displacement and in norm and norm for stress are derived, respectively. Numerical experiments are presented to confirm theoretical results.

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 Compact High Order Space-Time Method for Conservation Laws

AbstractThis paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L 2(L 2)”- and “DG”-norm formed by the “L 2(H 1)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.

Space–time discontinuous Galerkin finite element method for two-fluid flows

A novel numerical method for two-fluid flow computations is presented, which combines the space–time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space–time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local hp-refinement. The front tracking is incorporated via cut-cell mesh refinement to ensure a sharp interface between the fluids. To compute the interface dynamics the level set method (LSM) is used because of its ability to deal with merging and breakup. Also, the LSM is easy to extend to higher dimensions. Small cells arising from the cut-cell refinement are merged to improve the stability and performance. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative. The numerical method is applied to one and two dimensional two-fluid test problems using the Euler equations.

ADER Schemes for Nonlinear Systems of Stiff Advection–Diffusion–Reaction Equations

In this article we extend the high order ADER finite volume schemes introduced for stiff hyperbolic balance laws by Dumbser, Enaux and Toro (J. Comput. Phys. 227:3971---4001, 2008) to nonlinear systems of advection---diffusion---reaction equations with stiff algebraic source terms. We derive a new efficient formulation of the local space-time discontinuous Galerkin predictor using a nodal approach whose interpolation points are tensor-products of Gauss---Legendre quadrature points. Furthermore, we propose a new simple and efficient strategy to compute the initial guess of the locally implicit space-time DG scheme: the Gauss---Legendre points are initialized sequentially in time by a second order accurate MUSCL-type approach for the flux term combined with a Crank---Nicholson method for the stiff source terms. We provide numerical evidence that when starting with this initial guess, the final iterative scheme for the solution of the nonlinear algebraic equations of the local space-time DG predictor method becomes more efficient. We apply our new numerical method to some systems of advection---diffusion---reaction equations with particular emphasis on the asymptotic preserving property for linear model systems and the compressible Navier---Stokes equations with chemical reactions.

A priori error estimates of an extrapolated space-time discontinuous galerkin method for nonlinear convection-diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection-diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L∞(L2) –norm and the L2(H1) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010

Interactive High-Quality Visualization of Higher-Order Finite Elements

Higher-order finite element methods have emerged as an important discretization scheme for simulation. They are increasingly used in contemporary numerical solvers, generating a new class of data that must be analyzed by scientists and engineers. Currently available visualization tools for this type of data are either batch oriented or limited to certain cell types and polynomial degrees. Other approaches approximate higher-order data by resampling resulting in trade-offs in interactivity and quality. To overcome these limitations, we have developed a distributed visualization system which allows for interactive exploration of non-conforming unstructured grids, resulting from space-time discontinuous Galerkin simulations, in which each cell has its own higher-order polynomial solution. Our system employs GPU-based raycasting for direct volume rendering of complex grids which feature non-convex, curvilinear cells with varying polynomial degree. Frequency-based adaptive sampling accounts for the high variations along rays. For distribution across a GPU cluster, the initial object-space partitioning is determined by cell characteristics like the polynomial degree and is adapted at runtime by a load balancing mechanism. The performance and utility of our system is evaluated for different aeroacoustic simulations involving the propagation of shock fronts.

Adaptive spacetime method using Riemann jump conditions for coupled atomistic–continuum dynamics

We combine the Spacetime Discontinuous Galerkin (SDG) method for elastodynamics with the mathematically consistent Atomistic Discontinuous Galerkin (ADG) method in a new scheme that concurrently couples continuum and atomistic models of dynamic response in solids. The formulation couples non-overlapping continuum and atomistic models across sharp interfaces by weakly enforcing jump conditions, for both momentum balance and kinematic compatibility, using Riemann values to preserve the characteristic structure of the underlying hyperbolic system. Momentum balances to within machine-precision accuracy over every element, on each atom, and over the coupled system, with small, controllable energy dissipation in the continuum region that ensures numerical stability. When implemented on suitable unstructured spacetime grids, the continuum SDG model offers linear computational complexity in the number of elements and powerful adaptive analysis capabilities that readily bridge between atomic and continuum scales in both space and time. A special trace operator for the atomic velocities and an associated atomistic traction field enter the jump conditions at the coupling interface. The trace operator depends on parameters that specify, at the scale of the atomic spacing, the position of the coupling interface relative to the atoms. In a key finding, we demonstrate that optimizing these parameters suppresses spurious reflections at the coupling interface without the use of non-physical damping or special boundary conditions. We formulate the implicit SDG–ADG coupling scheme in up to three spatial dimensions, and describe an efficient iterative solution scheme that outperforms common explicit schemes, such as the Velocity Verlet integrator. Numerical examples, in 1 d × time and employing both linear and nonlinear potentials, demonstrate the performance of the SDG–ADG method and show how adaptive spacetime meshing reconciles disparate time steps and resolves atomic-scale signals in the continuum.

Multi-field spacetime discontinuous Galerkin methods for linearized elastodynamics

We extend the single-field spacetime discontinuous Galerkin (SDG) method for linearized elastodynamics of Abedi et al. [1] to multi-field versions. A three-field method, in displacement, velocity and strain, is derived by invoking a Bubnov–Galerkin weighted residuals procedure on the system of spacetime field equations and the corresponding jump conditions. A two-field formulation, in displacement and velocity, and the one-field displacement formulation of [1] are obtained from the three-field model through strong enforcement of kinematic compatibility relations. All of these formulations balance linear and angular momentum at the element level, and we prove that they are energy-dissipative and unconditionally stable. As in [1], we implement the SDG models using a causal, advancing-front meshing procedure that enables a patch-by-patch solution procedure with linear complexity in the number of spacetime elements. Numerical results show that the three-field formulation is most efficient, wherein all interpolated fields converge at the optimal, O ( h p + 1 ) , rate. For a given mesh size, the three-field model delivers error values that are more than an order of magnitude smaller than those of the one- and two-field models. The three-field formulation’s efficiency is also superior, independent of whether the comparison is based on matching polynomial orders or matching convergence rates.

An adaptive spacetime discontinuous Galerkin method for cohesive models of elastodynamic fracture

AbstractThis paper describes an adaptive numerical framework for cohesive fracture models based on a spacetime discontinuous Galerkin (SDG) method for elastodynamics with elementwise momentum balance. Discontinuous basis functions and jump conditions written with respect to target traction values simplify the implementation of cohesive traction–separation laws in the SDG framework; no special cohesive elements or other algorithmic devices are required. We use unstructured spacetime grids in a h‐adaptive implementation to adjust simultaneously the spatial and temporal resolutions. Two independent error indicators drive the adaptive refinement. One is a dissipation‐based indicator that controls the accuracy of the solution in the bulk material; the second ensures the accuracy of the discrete rendering of the cohesive law. Applications of the SDG cohesive model to elastodynamic fracture demonstrate the effectiveness of the proposed method and reveal a new solution feature: an unexpected quasi‐singular structure in the velocity response. Numerical examples demonstrate the use of adaptive analysis methods in resolving this structure, as well as its importance in reliable predictions of fracture kinetics. Copyright © 2009 John Wiley & Sons, Ltd.

Wave propagation in functionally graded and layered materials

In this study, wave propagation in functionally graded materials (FGMs) and layered materials is analyzed using space–time discontinuous Galerkin method introduced earlier by the authors [H.G. Aksoy, E. Senocak, Space–time discontinuous Galerkin method for dynamics of solids, Communications in Numerical Methods in Engineering 24 (2008) 1887–1907]. The numerical performance of the method is presented via two dimensional example problems of wave propagations. The effect of loading (quasi-static and dynamic) on the analysis of displacements and stresses is demonstrated. The results show that space–time discontinuous Galerkin method is successful in modelling wave propagation in FGMs and layered materials.

A Space-Time Expansion Discontinuous Galerkin Scheme With Local Time Stepping for the Ideal and Viscous MHD Equations

In this paper, we present the extension of the space-time expansion discontinuous Galerkin to handle ideal and viscous magnetohydrodynamics (MHD) equations. The local time-stepping strategy that this scheme is capable of allows each cell to have its own time step whereas the high order of accuracy in time is retained. This may significantly speed up calculations. The diffusive flux is evaluated through a so-called diffusive generalized Riemann problem. The divergence constraint of the MHD equations is addressed, and a hyperbolic cleaning method is shown that can be enhanced by utilizing the local time-stepping framework. MHD problems such as the Orszag-Tang vortex or the magnetic blast problem are performed to challenge the capabilities of the proposed space-time expansion scheme.

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.

Generalized EVI‐space‐time Galerkin method for dynamical modeling in porous media

AbstractWe investigate a generalized space–time discontinuous Galerkin formulation for modeling dynamical phenomena in porous media. The finite element approximation is based on a coupled space–time discretization. An advanced Embedded Velocity Integration (EVI) technique is applied to circumvent the direction solution of the displacement fields but to solve the rate term of the unknowns. Moverover, a stabilization factor α is introduced to modify the integration scheme of the velocities, which further contributes to the numerical stabilization of the overall solution. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.