New local discontinuous Galerkin method for a fractional time diffusion wave equation

ABSTRACT

This paper is devoted to the study of a new discontinuous finite element idea for the time fractional diffusion-wave equation defined in bounded domain. The time fractional derivatives are described in the Caputo's sense. By applying the sine transform on the time fractional diffusion-wave equation, we make the equation depend on time. Then we use definition of Caputo's derivative and by defining l-degree discontinuous finite element with interpolated coefficients we solve the mentioned equation. Error estimate, existence and uniqueness are proved. Finally, the theoretical results are tested by some numerical examples.     

Three dimensional discontinuous Galerkin methods for Euler equations on adaptive conforming meshes

In the numerical simulation of three dimensional fluid dynamical equations, the huge computational quantity is a main challenge. In this paper, the discontinuous Galerkin (DG) finite element method combined with the adaptive mesh refinement (AMR) is studied to solve the three dimensional Euler equations based on conforming unstructured tetrahedron meshes, that is according the equation solution variation to refine and coarsen grids so as to decrease total mesh number. The four space adaptive strategies are given and analyzed their advantages and disadvantages. The numerical examples show the validity of our methods.     

Stabilized discontinuous Galerkin method for hyperbolic equations

In this work a new stabilization technique is proposed and studied for the discontinuous Galerkin method applied to hyperbolic equations. In order to avoid the use of slope limiters, a streamline diffusion-like term is added to control oscillations for arbitrary element orders. Thus, the scheme combines ideas from both the Runge-Kutta discontinuous Galerkin method [J. Scient. Comput. 16 (2001) 173] and the streamline diffusion method [Comput. Methods Appl. Mech. Engrg. 32 (1982)]. To increase the stability range of the method, the diffusion term is treated implicitly. The result is a scheme with higher order in space with the same stability range as the finite volume method. An optimal relation between the time step and the size of the diffusion coefficient is analyzed for numerical precision. The scheme is implemented using the object oriented programming philosophy based on the environment described in [Comput. Methods Appl. Mech. Engrg. 150 (1997)]. Accuracy and shock capturing abilities of the method are analyzed in terms of two bidimensional model problems: the rotating cone and the backward facing step problem for the Euler equations of gas dynamics.     

Three time integration methods for incompressible flows with discontinuous Galerkin Boltzmann method

This paper presents three time integration methods for incompressible flows with finite element method in solving the lattice-BGK Boltzmann equation. The space discretization is performed using nodal discontinuous Galerkin method, which employs unstructured meshes with triangular elements and high order approximation degrees. The time discretization is performed using three different kinds of time integration methods, namely, direct, decoupling and splitting. From the storage cost, temporal accuracy, numerical stability and time consumption, we systematically compare three time integration methods. Then benchmark fluid flow simulations are performed to highlight efficient time integration methods. Numerical results are in good agreement with others or exact solutions.     

Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations

We consider the approximation of a simplified model of the depth-averaged two-dimensional shallow water equations by two approaches. In both approaches, a discontinuous Galerkin (DG) method is used to approximate the continuity equation. In the first approach, a continuous Galerkin method is used for the momentum equations. In the second approach a particular DG method, the nonsymmetric interior penalty Galerkin method, is used to approximate momentum. A priori error estimates are derived and numerical results are presented for both approaches.     

A discontinuous Galerkin method for the wave equation

We present a new discontinuous Galerkin method for solving the second-order wave equation using the standard continuous finite element method in space and a discontinuous method in time directly applied to second-order ode systems. We prove several optimal a priori error estimates in space–time norms for this new method and show that it can be more efficient than existing methods. We also write the leading term of the local discretization error in terms of Lobatto polynomials in space and Jacobi polynomials in time which leads to superconvergence points on each space–time cell. We discuss how to apply our results to construct efficient and asymptotically exact a posteriori estimates for space–time discretization errors. Numerical results are in agreement with theory.     

A Runge-Kutta discontinuous Galerkin method for the Euler equations

This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for the Euler equations of gas dynamics from the viewpoint of kinetic theory. Like the traditional gas-kinetic schemes, our proposed RKDG method does not need to use the characteristic decomposition or the Riemann solver in computing the numerical flux at the surface of the finite elements. The integral term containing the non-linear flux can be computed exactly at the microscopic level. A limiting procedure is carefully designed to suppress numerical oscillations. It is demonstrated by the numerical experiments that the proposed RKDG methods give higher resolution in solving problems with smooth solutions. Moreover, shock and contact discontinuities can be well captured by using the proposed methods.     

A discontinuous Galerkin algorithm for the two-dimensional shallow water equations

A new high-resolution finite element scheme is introduced for solving the two-dimensional (2D) depth-integrated shallow water equations (SWE) via local plane approximations to the unknowns. Bed topography data are locally approximated in the same way as the flow variables to render an instinctive well-balanced scheme. A finite volume (FV) wetting and drying technique that reconstructs the Riemann states by ensuring non-negative water depth and maintaining well-balanced solution is adjusted and implemented in the current finite element framework. Meanwhile, a local slope-limiting process is applied and those troubled-slope-components are restricted by the minmod FV slope limiter. The inter-cell fluxes are upwinded using the HLLC approximate Riemann solver. Friction forces are separately evaluated via stable implicit discretization to the finite element approximating coefficients. Boundary conditions are derived and reported in details. The present model is validated against several test cases including dam-break flows on regular and irregular domains with flooding and drying.     

Coupling of continuous and discontinuous Galerkin methods for transport problems

In this paper, we formulate a coupled discontinuous/continuous Galerkin method for the numerical solution of convection–diffusion (transport) equations, where convection may be dominant. One motivation for this approach is to use a discontinuous method where the solution is rough, e.g., in regions of high gradients, and use a continuous method where the solution is smooth. In this approach, the domain is decomposed into two regions, and appropriate transmission conditions are applied at the interface between regions. In one region, a local discontinuous Galerkin method is applied, and in the other region a standard continuous Galerkin method is employed. Stability and a priori error estimates for the coupled method are derived, and numerical results in one space dimension are given for smooth problems and problems with sharp fronts.     

Multilevel methods for discontinuous Galerkin FEM on locally refined meshes

We review shortly the analytical results on multilevel preconditioning for DG methods for elliptic problems. An algorithm is presented which does not loose efficiency under local refinement. It uses subspace smoothing and does not require integration of additional matrices. This algorithm is applied to the interior penalty method and the local discontinuous Galerkin method. A feasible way of implementing the scheme is presented.     

Error analysis of a two-grid discontinuous Galerkin method for non-linear parabolic equations

Discontinuous Galerkin (DG) approximations for non-linear parabolic problems are investigated. To linearize the discretized equations, we use a two-grid method involving a small non-linear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates in H¹-norm are obtained, O(h^r+H^r+1) where r is the order of the DG space. The analysis shows that our two-grid DG algorithm will achieve asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H^(r+1)/r). The numerical experiments verify the efficiency of our algorithm.     

Expression templates implementation of continuous and discontinuous Galerkin methods

Efficiency and flexibility are often mutually exclusive features in a code. This still prompts a large part of the Scientific Computing community to use traditional procedural languages. In the last years, however, new programming techniques have been introduced allowing for a high level of abstraction without loss of performance. In this paper we present an application of the Expression Templates technique introduced in (Veldhuizen in Expression templates. C++ Report magazine, vol 7, pp 26–31, 1995) to the assembly step of a finite element computation. We show that a suitable implementation, such that the compiler has the role of parsing abstract operations, allows for user-friendliness. Moreover, it gains in performance with respect to more traditional techniques for achieving this kind of abstraction. Both the cases of conforming and discontinuous Galerkin finite element discretization are considered. The proposed implementation is finally applied to a number of problems entailing different kind of complications.     

Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems

An a priori error analysis of discontinuous Galerkin methods for a general elliptic problem is derived under a mild elliptic regularity assumption on the solution. This is accomplished by using some techniques from a posteriori error analysis. The model problem is assumed to satisfy a Gårding type inequality. Optimal order L ₂ norm a priori error estimates are derived for an adjoint consistent interior penalty method.     

A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations

In this work we consider Runge–Kutta discontinuous Galerkin methods for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics.     

Numerical methods of Lagrangian points and particles for one-dimensional wave gas-dynamics equations

In this paper, an approach is formulated to construct numerical methods of computational gas dynamics based on the approximation of the second-order nonlinear wave equations (NWE) in time and spatial variables. The NWE approach enables the construction of finite-difference and finite-element schemes with the balance (conservation) cells both in the form of finite volumes and Lagrangian points and particles. That is why the numerical methods based on the NWE approximation are of great interest for the solution of both one-dimensional (1D) and multidimensional problems of gas dynamics. In this work, we construct and study discrete NWE models for 1D problems of gas dynamics in Lagrangian variables and discuss the results of numerical experiments.     

Fully computable bounds for a staggered discontinuous Galerkin method for the Stokes equations

We first propose a guaranteed upper bound for an arbitrary order staggered discontinuous Galerkin (staggered DG) method for the Stokes equations with the use of the global inf–sup constant. Equilibrated stress reconstruction and velocity reconstruction are the main ingredients in the construction of the error estimator. Next, to improve the error estimation and to overcome the difficulties caused by the calculation of the global inf–sup constant, a refined error control relying on local inf–sup constants is also developed. Some minimization techniques and an explicit method are then established to facilitate the construction of the refined error control. Finally, some benchmark examples are tested to compare the performances of the proposed error estimators.     

Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations

By incorporating the Legendre multiwavelet into the mixed discontinuous Galerkin method, in this paper, we present a novel method for solving second-order elliptic partial differential equations (PDEs), which is known as the mixed discontinuous Legendre multiwavelet Galerkin method, derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. The proposed method is also applicable to some other kinds of PDEs.