Table of Content

20 June 2022, Volume 4 Issue 2

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Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System

Jiawei Sun, Shusen Xie, Yulong Xing

2022, 4(2):  381-416.  doi:10.1007/s42967-021-00119-4

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Boussinesq type equations have been widely studied to model the surface water wave. In this paper, we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system, the BBM-BBM system, the Bona-Smith system, etc. We propose local discontinuous Galerkin (LDG) methods, with carefully chosen numerical fluxes, to numerically solve this abcd Boussinesq system. The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a, b, c, d. Numerical experiments are shown to test the convergence rates, and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.

Capitalizing on Superconvergence for More Accurate Multi-Resolution Discontinuous Galerkin Methods

Jennifer K. Ryan

2022, 4(2):  417-436.  doi:10.1007/s42967-021-00121-w

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This article focuses on exploiting superconvergence to obtain more accurate multi-resolution analysis. Specifcally, we concentrate on enhancing the quality of passing of information between scales by implementing the Smoothness-Increasing Accuracy-Conserving (SIAC) fltering combined with multi-wavelets. This allows for a more accurate approximation when passing information between meshes of diferent resolutions. Although this article presents the details of the SIAC flter using the standard discontinuous Galerkin method, these techniques are easily extendable to other types of data.

Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids

Mahboub Baccouch

2022, 4(2):  437-476.  doi:10.1007/s42967-021-00123-8

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This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin (LDG) method for two-dimensional semilinear second-order elliptic problems of the form -Δu=f (x, y, u) on Cartesian grids. By introducing special GaussRadau projections and using duality arguments, we obtain, under some suitable choice of numerical fuxes, the optimal convergence order in L²-norm of O(h^p+1) for the LDG solution and its gradient, when tensor product polynomials of degree at most p and grid size h are used. Moreover, we prove that the LDG solutions are superconvergent with an order p + 2 toward particular Gauss-Radau projections of the exact solutions. Finally, we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves (p + 1)-th order superconvergence. Some numerical experiments are performed to illustrate the theoretical results.

Superconvergent Interpolatory HDG Methods for Reaction Difusion Equations II: HHO-Inspired Methods

Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang

2022, 4(2):  477-499.  doi:10.1007/s42967-021-00128-3

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In Chen et al. (J. Sci. Comput. 81(3):2188-2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defned on simplicial meshes, for scalar reaction-difusion equations and showed how to defne an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree k ≥ 1. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3):635-650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defned on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements afne-equivalent to a fnite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for k ≥ 0 by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory.

A Discontinuous Galerkin Method for Blood Flow and Solute Transport in One-Dimensional Vessel Networks

Rami Masri, Charles Puelz, Beatrice Riviere

2022, 4(2):  500-529.  doi:10.1007/s42967-021-00126-5

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This paper formulates an efficient numerical method for solving the convection diffusion solute transport equations coupled to blood flow equations in vessel networks. The reduced coupled model describes the variations of vessel cross-sectional area, radially averaged blood momentum and solute concentration in large vessel networks. For the discretization of the reduced transport equation, we combine an interior penalty discontinuous Galerkin method in space with a novel locally implicit time stepping scheme. The stability and the convergence are proved. Numerical results show the impact of the choice for the steady-state axial velocity profile on the numerical solutions in a fifty-five vessel network with physiological boundary data.

Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Methods for KdV Type Equations

Xue Hong, Yinhua Xia

2022, 4(2):  530-562.  doi:10.1007/s42967-021-00127-4

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In this paper, several arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods are presented for Korteweg-de Vries (KdV) type equations on moving meshes. Based on the L² conservation law of KdV equations, we adopt the conservative and dissipative numerical fuxes for the nonlinear convection and linear dispersive terms, respectively. Thus, one conservative and three dissipative ALE-DG schemes are proposed for the equations. The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper. In addition, the L² -norm error estimates are also proved for two schemes, whose numerical fuxes for the linear dispersive term are both dissipative type. More precisely, when choosing the approximation space with the piecewise kth degree polynomials, the error estimate provides the kth order of convergence rate in L²-norm for the scheme with the conservative numerical fuxes applied for the nonlinear convection term. Furthermore, the (k + 1∕2) th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fuxes applied for the convection term. Moreover, a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations. Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.

Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations

Xiaobing Feng, Thomas Lewis, Aaron Rapp

2022, 4(2):  563-596.  doi:10.1007/s42967-021-00130-9

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This paper develops and analyzes a new family of dual-wind discontinuous Galerkin (DG) methods for stationary Hamilton-Jacobi equations and their vanishing viscosity regularizations. The new DG methods are designed using the DG fnite element discrete calculus framework of[17] that defnes discrete diferential operators to replace continuous differential operators when discretizing a partial diferential equation (PDE). The proposed methods, which are non-monotone, utilize a dual-winding methodology and a new skewsymmetric DG derivative operator that, when combined, eliminate the need for choosing indeterminable penalty constants. The relationship between these new methods and the local DG methods proposed in[38] for Hamilton-Jacobi equations as well as the generalized-monotone fnite diference methods proposed in[13] and corresponding DG methods proposed in[12] for fully nonlinear second order PDEs is also examined. Admissibility and stability are established for the proposed dual-wind DG methods. The stability results are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone schemes. Numerical experiments are provided to gauge the performance of the new methods.

Hybrid High-Order Methods for the Acoustic Wave Equation in the Time Domain

Erik Burman, Omar Duran, Alexandre Ern

2022, 4(2):  597-633.  doi:10.1007/s42967-021-00131-8

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We devise hybrid high-order (HHO) methods for the acoustic wave equation in the time domain. We frst consider the second-order formulation in time. Using the Newmark scheme for the temporal discretization, we show that the resulting HHO-Newmark scheme is energy-conservative, and this scheme is also amenable to static condensation at each time step. We then consider the formulation of the acoustic wave equation as a frst-order system together with singly-diagonally implicit and explicit Runge-Kutta (SDIRK and ERK) schemes. HHO-SDIRK schemes are amenable to static condensation at each time step. For HHO-ERK schemes, the use of the mixed-order formulation, where the polynomial degree of the cell unknowns is one order higher than that of the face unknowns, is key to beneft from the explicit structure of the scheme. Numerical results on test cases with analytical solutions show that the methods can deliver optimal convergence rates for smooth solutions of order O(h^k+1) in the H¹-norm and of order O(h^k+2) in the L²-norm. Moreover, test cases on wave propagation in heterogeneous media indicate the benefts of using high-order methods.

Discontinuous Galerkin Methods for a Class of Nonvariational Problems

Andreas Dedner, Tristan Pryer

2022, 4(2):  634-656.  doi:10.1007/s42967-021-00133-6

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We extend the fnite element method introduced by Lakkis and Pryer (SIAM J. Sci. Comput. 33(2):786-801, 2011) to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the "fnite element Hessian" as an auxiliary variable in the formulation. Representing the fnite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble; thus, this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach. We conduct a stability and consistency analysis making use of the unifed frameworkset out in Arnold et al. (SIAM J. Numer. Anal. 39(5):1749-1779, 2001/2002). We also give an a posteriori analysis of the method in the case where the problem has a strong solution. The analysis applies to any consistent representation of the fnite element Hessian, and thus is applicable to the previous works making use of continuous Galerkin approximations. Numerical evidence is presented showing that the method works well also in a more general setting.

Extendible and Efcient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods

Andreas Dedner, Robert Klöfkorn

2022, 4(2):  657-696.  doi:10.1007/s42967-021-00134-5

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This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efcient implementations of the discontinuous Galerkin (DG) method for solving a wide range of nonlinear partial diferential equations (PDEs). Although the C++ interfaces of Dune-Fem-DG are highly fexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work, easier user interfaces based on Python and the unifed form language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and frst-order hyperbolic PDEs.

Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refnement

Will Pazner, Tzanio Kolev

2022, 4(2):  697-727.  doi:10.1007/s42967-021-00136-3

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In this paper, we develop subspace correction preconditioners for discontinuous Galerkin (DG) discretizations of elliptic problems with hp-refnement. These preconditioners are based on the decomposition of the DG fnite element space into a conforming subspace, and a set of small nonconforming edge spaces. The conforming subspace is preconditioned using a matrix-free low-order refned technique, which in this work, we extend to the hp-refnement context using a variational restriction approach. The condition number of the resulting linear system is independent of the granularity of the mesh h, and the degree of the polynomial approximation p. The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees. Numerical examples are shown on several test cases involving adaptively and randomly refned meshes, using both the symmetric interior penalty method and the second method of Bassi and Rebay (BR2).

Development of a Balanced Adaptive Time-Stepping Strategy Based on an Implicit JFNK-DG Compressible Flow Solver

Yu Pan, Zhen-Guo Yan, Joaquim Peiró, Spencer J. Sherwin

2022, 4(2):  728-757.  doi:10.1007/s42967-021-00138-1

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A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations. A proper relation between the spatial, temporal and iterative errors generated within one time step is constructed. With an estimate of temporal and spatial error using an embedded RungeKutta scheme and a higher order spatial discretization, an adaptive time-stepping strategy is proposed based on the idea that the time step should be the maximum without obviously infuencing the total error of the discretization. The designed adaptive time-stepping strategy is then tested in various types of problems including isentropic vortex convection, steady-state fow past a fat plate, Taylor-Green vortex and turbulent fow over a circular cylinder at Re=3 900. The results indicate that the adaptive time-stepping strategy can maintain that the discretization error is dominated by the spatial error and relatively high efciency is obtained for unsteady and steady, well-resolved and under-resolved simulations.