An Adaptive Moving Mesh Discontinuous Galerkin Method for the Radiative Transfer Equation

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, frequency, space and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are demonstrated in a selection of one- and two-dimensional numerical examples.     

A GPU-Accelerated Hybridizable Discontinuous Galerkin Method for Linear Elasticity

We design and analyze an efficient GPU-accelerated hybridizable discontinuous Galerkin method for linear elasticity. Performance analysis of the method is done using the state-of-the-art Time-Accuracy-Size (TAS) spectrum. TAS is a new performance measure which takes into account the accuracy of the solution. Standard performance measures, like floating point operations or run-time, are not completely appropriate for gauging the performance of approximations of continuum mechanics problems, as they neglect the solutions accuracy. A standard roofline model demonstrates that our method is utilizing computational resources efficiently, and as such, significant speed ups over a serial implementation are obtained. By combining traditional performance measures and the novel time-accuracy measures [7] into our performance model, we are able to draw more complete conclusions about which discretizations are best suited for an application. Several numerical experiments validate and verify our numerical scheme.     

A Weighted Runge-Kutta Discontinuous Galerkin Method for 3D Acoustic and Elastic Wave-Field Modeling

Numerically solving 3D seismic wave equations is a key requirement for forward modeling and inversion. Here, we propose a weighted Runge-Kutta discontinuous Galerkin (WRKDG) method for 3D acoustic and elastic wave-field modeling. For this method, the second-order seismic wave equations in 3D heterogeneous anisotropic media are transformed into a first-order hyperbolic system, and then we use a discontinuous Galerkin (DG) solver based on numerical-flux formulations for spatial discretization. The time discretization is based on an implicit diagonal Runge-Kutta (RK) method and an explicit iterative technique, which avoids solving a large-scale system of linear equations. In the iterative process, we introduce a weighting factor. We investigate the numerical stability criteria of the 3D method in detail for linear and quadratic spatial basis functions. We also present a 3D analysis of numerical dispersion for the full discrete approximation of acoustic equation, which demonstrates that the WRKDG method can efficiently suppress numerical dispersion on coarse grids. Numerical results for several different 3D models including homogeneous and heterogeneous media with isotropic and anisotropic cases show that the 3D WRKDG method can effectively suppress numerical dispersion and provide accurate wave-field information on coarse mesh.     

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

In this paper, a new multi-resolution weighted essentially non-oscillatory (MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENO limiter is a new extension of the finite volume MR-WENO schemes. Such new limiter uses information of the LDG solution essentially only within the troubled cell itself, to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the highest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on condition that their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the freedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiter is very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classicalviscous examples are given to show the good performance of this third-order LDG method with associated MR-WENO limiter.     

A High-Order Cell-Centered Discontinuous Galerkin Multi-Material Arbitrary Lagrangian-Eulerian Method

In this paper, a high-order cell-centered discontinuous Galerkin (DG) multi-material arbitrary Lagrangian-Eulerian (MMALE) method is developed for compressible fluid dynamics. The MMALE method utilizes moment-of-fluid (MOF) interface reconstruction technology to simulate multi-materials of immiscible fluids. It is an explicit time-marching Lagrangian plus remap type. In the Lagrangian phase, an updated high-order discontinuous Galerkin Lagrangian method is applied for the discretization of hydrodynamic equations, and Tipton's pressure relaxation closure model is used in the mixed cells. A robust moment-of-fluid interface reconstruction algorithm is used to provide the information of the material interfaces for remapping. In the rezoning phase, Knupp's algorithm is used for mesh smoothing. For the remapping phase, a high-order accurate remapping method of the cell-intersection-based type is proposed. It can be divided into four stages: polynomial reconstruction, polygon intersection, integration, and detection of problematic cells and limiting. Polygon intersection is based on the "clipping and projecting" algorithm, and detection of problematic cells depends on a troubled cell marker, and a posteriori multi-dimensional optimal order detection (MOOD) limiting strategy is used for limiting. Numerical tests are given to demonstrate the robustness and accuracy of our method.     

Discrete Duality Finite Volume Discretization of the Thermal-$P_N$ Radiative Transfer Equations on General Meshes

The discrete duality finite volume method has proven to be a practical tool for discretizing partial differential equations coming from a wide variety of areas of physics on nearly arbitrary meshes. The main ingredients of the method are: (1) use of three meshes, (2) use of the Gauss-Green theorem for the approximation of derivatives, (3) discrete integration by parts. In this article we propose to extend this method to the coupled grey thermal-$P_N$ radiative transfer equations in Cartesian and cylindrical coordinates in order to be able to deal with two-dimensional Lagrangian approximations of the interaction of matter with radiation. The stability under a Courant-Friedrichs-Lewy condition and the preservation of the diffusion asymptotic limit are proved while the experimental second-order accuracy is observed with manufactured solutions. Several numerical experiments are reported which show the good behavior of the method.     

Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation

In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in one dimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do not need numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L²error estimates are established for the linear and nonlinear equation using several projections for different parameter choices. Although we can not provide optimal a priori error estimate, numerical examples show that our scheme attains the optimal (k+1)-th order of accuracy when using piecewise k-th degree polynomials for many cases.     

A Study of Several Artificial Viscosity Models within the Discontinuous Galerkin Framework

Dealing with strong shocks while retaining low numerical dissipation traditionally has been one of the major challenges for high order methods like discontinuous Galerkin (DG). In the literature, shock capturing models have been designed for DG based on various approaches, such as slope limiting, (H)WENO reconstruction, a posteriori sub-cell limiting, and artificial viscosity, among which a subclass of artificial viscosity methods are compared in the present work. Four models are evaluated, including a dilation-based model, a highest modal decay model, an averaged modal decay model, and an entropy viscosity model. Performance for smooth, non-smooth and broadband problems are examined with typical one- and two-dimensional cases.     

High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes

In this paper, the second-order and third-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO limiter is an extension of a finite volume multi-resolution WENO scheme developed in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and third-order RKDG methods with the associated multi-resolution WENO limiters are developed as examples for general high-order RKDG methods, which could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property near strong discontinuities by gradually degrading from the optimal order to the first order. The linear weights inside the procedure of the new multi-resolution WENO limiters can be set as any positive numbers on the condition that they sum to one. A series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell on tetrahedral meshes. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such spatial reconstruction methodology improves the robustness in the simulation on the same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes) and three-dimensional tests are performed to demonstrate the good performance of the RKDG methods with new multi-resolution WENO limiters.     

Three Discontinuous Galerkin Methods for One- and Two-Dimensional Nonlinear Dirac Equations with a Scalar Self-Interaction

This paper develops three high-order accurate discontinuous Galerkin (DG) methods for the one-dimensional (1D) and two-dimensional (2D) nonlinear Dirac (NLD) equations with a general scalar self-interaction. They are the Runge-Kutta DG (RKDG) method and the DG methods with the one-stage fourth-order Lax-Wendroff type time discretization (LWDG) and the two-stage fourth-order accurate time discretization (TSDG). The RKDG method uses the spatial DG approximation to discretize the NLD equations and then utilize the explicit multistage high-order Runge-Kutta time discretization for the first-order time derivatives, while the LWDG and TSDG methods, on the contrary, first give the one-stage fourth-order Lax-Wendroff type and the two-stage fourth-order time discretizations of the NLD equations, respectively, and then discretize the first- and higher-order spatial derivatives by using the spatial DG approximation. The $L^2$ stability of the 2D semi-discrete DG approximation is proved in the RKDG methods for a general triangulation, and the computational complexities of three 1D DG methods are estimated. Numerical experiments are conducted to validate the accuracy and the conservation properties of the proposed methods. The interactions of the solitary waves, the standing and travelling waves are investigated numerically and the 2D breathing pattern is observed.     

A Bilinear Immersed Finite Volume Element Method for the Diffusion Equation with Discontinuous Coefficient

This paper is to present a finite volume element (FVE) method based on the bilinear immersed finite element (IFE) for solving the boundary value problems of the diffusion equation with a discontinuous coefficient (interface problem). This method possesses the usual FVE method's local conservation property and can use a structured mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient has discontinuity along piecewise smooth nontrivial curves. Numerical examples are provided to demonstrate features of this method. In particular, this method can produce a numerical solution to an interface problem with the usual O(h²) (in L² norm) and O(h) (in H¹ norm) convergence rates.     

Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation

This study aimed to specialise a directional $\mathcal{H}^2 (\mathcal{D}\mathcal{H}^2)$ compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal $\mathcal{D}\mathcal{H}^2$ approximation for low and high-frequency problems. We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a $\mathcal{D}\mathcal{H}^2$ gives better computational efficiency than a classical BEM in the case of high-order finite elements and $hp$ heterogeneous meshes. The results indicate that DG is suitable for an auto-adaptive context in integral equations.     

Dissipative and Conservative Local Discontinuous Galerkin Methods for the Fornberg-Whitham Type Equations

In this paper, we construct high order energy dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations. We give the proofs for the dissipation and conservation for related conservative quantities. The corresponding error estimates are proved for the proposed schemes. The capability of our schemes for different types of solutions is shown via several numerical experiments. The dissipative schemes have good behavior for shock solutions, while for a long time approximation, the conservative schemes can reduce the shape error and the decay of amplitude significantly.     

Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme

We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on identifying the correct endpoints of this interval. The first-order optimality conditions from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward schemes with the Runge-Kutta method. To illustrate the feasibility of the approach, we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters of the random variable in the uncertain differential equation even if discontinuities are present.     

A CFD-Aided Galerkin Method for Global Linear Instability Analysis

Global linear instability analysis is a powerful tool for the complex flow diagnosis. However, the methods used in the past would generally suffer from some disadvantages, either the excessive computational resources for the low-order methods or the tedious mathematical derivations for the high-order methods. The present work proposed a CFD-aided Galerkin methodology which combines the merits from both the low-order and high-order methods, where the expansion on proper basis functions is preserved to ensure a small matrix size, while the differentials, incompressibility constraints and boundary conditions are realized by applying the low-order linearized Navier-Stokes equation solvers on the basis functions on a fine grid. Several test cases have shown that the new method can get satisfactory results for one-dimensional, two-dimensional and three-dimensional flow problems and also for the problems with complex geometries and boundary conditions.     

A Diagonal Sweeping Domain Decomposition Method with Source Transfer for the Helmholtz Equation

In this paper, we propose and test a novel diagonal sweeping domain decomposition method (DDM) with source transfer for solving the high-frequency Helmholtz equation in$\mathbb{R}^n$. In the method the computational domain is partitioned into overlapping checkerboard subdomains for source transfer with the perfectly matched layer (PML) technique, then a set of diagonal sweeps over the subdomains are specially designed to solve the system efficiently. The method improves the additive overlapping DDM [43] and the L-sweeps method [50] by employing a more efficient subdomain solving order. We show that the method achieves the exact solution of the global PML problem with $2^n$ sweeps in the constant medium case. Although the sweeping usually implies sequential subdomain solves, the number of sequential steps required for each sweep in the method is only proportional to the $n$-th root of the number of subdomains when the domain decomposition is quasi-uniform with respect to all directions, thus it is very suitable for parallel computing of the Helmholtz problem with multiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to demonstrate the effectiveness and efficiency of the proposed method.     

FESTUNG: A MATLAB / GNU Octave Toolbox for the Discontinuous Galerkin Method. Part IV: Generic Problem Framework and Model-Coupling Interface

This is the fourth installment in our series on implementing the discontinuous Galerkin (DG) method as an open source MATLAB / GNU Octave toolbox. Similarly to its predecessors, this part presents new features for application developers employing DG methods and follows our strategy of relying on fully vectorized constructs and supplying a comprehensive documentation. The specific focus of the current work is the newly added generic problem implementation framework and the highly customizable model-coupling interface for multi-domain and multi-physics simulation tools based on this framework. The functionality of the coupling interface in the FESTUNG toolbox is illustrated using a two-way coupled free-surface / groundwater flow system as an example application.     

A Discrete-Ordinate Discontinuous-Streamline Diffusion Method for the Radiative Transfer Equation

The radiative transfer equation (RTE) arises in many different areas of science and engineering. In this paper, we propose and investigate a discrete-ordinate discontinuous-streamline diffusion (DODSD) method for solving the RTE, which is a combination of the discrete-ordinate technique and the discontinuous-streamline diffusion method. Different from the discrete-ordinate discontinuous Galerkin (DODG) method for the RTE, an artificial diffusion parameter is added to the test functions in the spatial discretization. Stability and error estimates in certain norms are proved. Numerical results show that the proposed method can lead to a more accurate approximation in comparison with the DODG method.     

An Efficient DDG/FV Hybrid Method for 3D Viscous Flow Simulations on Tetrahedral Grids

A new hybrid reconstruction scheme DDG/FV is developed in this work by combining the DDG method and DG/FV hybrid scheme developed in the authors' previous work [1–4] to simulate three-dimensional compressible viscous flow on tetrahedral grids. The extended von Neumann stencils are used in the reconstruction process to ensure the linear stability, and the L₂ projection and the least-squares method are adopted to reconstruct higher order distributions for higher accuracy and robustness. In addition, a quadrature-free L₂ projection based on orthogonal basis functions is implemented to improve the efficiency of reconstruction. Three typical test cases, including the 3D Couette flow, laminar flows over an analytical 3D body of revolution and over a sphere, are simulated to validate the accuracy and efficiency of DDG/FV method. The numerical results demonstrate that the DDG scheme can accelerate the convergence history compared with widely-used BR2 scheme. More attractively, the new DDG/FV hybrid method can deliver the same accuracy as BR2-DG method with more than 2 times of efficiency improvement in solving 3D Navier-Stokes equations on tetrahedral grids, and even one-order of magnitude faster in some cases, which shows good potential in future realistic applications.     

An LDG Method for Stochastic Cahn-Hilliard Type Equation Driven by General Multiplicative Noise Involving Second-Order Derivative

In this paper, we propose a local discontinuous Galerkin (LDG) method for the multi-dimensional stochastic Cahn-Hilliard type equation in a general form, which involves second-order derivative $∆u$ in the multiplicative noise. The stability of our scheme is proved for arbitrary polygonal domain with triangular meshes. We get the sub-optimal error estimate $\mathbb{O}(h^k)$ if the Cartesian meshes with $Q^k$ elements are used. Numerical examples are given to display the performance of the LDG method.