Superconvergence of Direct Discontinuous Galerkin Methods:Eigen-structure Analysis Based on Fourier Approach

This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.

The Error Estimates of Direct Discontinuous Galerkin Methods Based on Upwind-Baised Fluxes

In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the optimal error estimates. The numerical experiments imply that L2 norms error estimates can reach to order k + 1 by using time discretization methods.

The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Diffusion Problems

In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear convection-diffusion problems.In the spatial discretization,both the original DDG methods and the refined DDG methods with interface corrections are considered.In the time discretization,the convection term is treated explicitly and the diffusion term implicitly.By the energy method,we show that the corresponding fully discrete schemes are unconditionally stable,in the sense that the time-step is only required to be upper bounded by a constant which is independent of the mesh size h.Opti-mal error estimate is also obtained by the aid of a special global projection.Numerical experiments are given to verify the stability and accuracy of the proposed schemes.

Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations

In this paper,we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin(DDG)method(Liu and Yan in SIAM J Numer Anal 47(1):475-698,2009),the DDG method with the interface correction(DDGIC)(Liu and Yan in Commun Comput Phys 8(3):541-564,2010),the symmetric DDG method(Vidden and Yan in Comput Math 31(6):638-662,2013),and the nonsymmetric DDG method(Yan in J Sci Comput 54(2):663-683,2013).We also include the study of the interior penalty DG(IPDG)method,due to its close relation to DDG methods.Error estimates are carried out for both P2 and P3 polynomial approximations.By investigating the quantitative errors at the Lobatto points,we show that the DDGIC and symmetric DDG methods are superior,in the sense of obtaining(k+2)th superconvergence orders for both P2 and P3 approximations.Superconvergence order of(k+2)is also observed for the IPDG method with P3 polynomial approximations.The errors are sensitive to the choice of the numerical flux coefficient for even degree P2 approximations,but are not for odd degree P3 approxi-mations.Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.

Direct discontinuous Galerkin method for the generalized Burgers-Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge^Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.

Solving coupled nonlinear Schrodinger equations via a direct discontinuous Galerkin method

In this work, we present the direct discontinuous Galerkin （DDG） method for the one-dimensional coupled nonlinear Schr5dinger （CNLS） equation. We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system. The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge Kutta method. Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.

Solving coupled nonlinear Schrödinger equations via a direct discontinuous Galerkin method

In this work,we present the direct discontinuous Galerkin(DDG) method for the one-dimensional coupled nonlinear Schrdinger(CNLS) equation.We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system.The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method.Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.

A High-Order Direct Discontinuous Galerkin Method for Variable Density Incompressible Flows

In this work,we develop a novel high-order discontinuous Galerkin(DG)method for solving the incompressible Navier-Stokes equations with variable density.The incompressibility constraint at cell interfaces is relaxed by an artificial compressibility term.Then,since the hyperbolic nature of the governing equations is recovered,the simple and robust Harten-Lax-van Leer(HLL)flux is applied to discrete the inviscid term of the variable density incompressible Navier-Stokes equations.The viscous term is discretized by the direct DG(DDG)method,the construction of which was initially inspired by the weak solution of a scalar diffusion equation.In addition,in order to eliminate the spurious oscillations around sharp density gradients,a local slope limiting operator is also applied during the highly stratified flow simulations.The convergence property and performance of the present high-order DDG method are well demonstrated by several benchmark and challenging numerical test cases.Due to its advantages of simplicity and robustness in implementation,the present method offers an effective approach for simulating the variable density incompressible flows.

A Direct Discontinuous Galerkin Method with Interface Correction for the Compressible Navier-Stokes Equations on Unstructured Grids

Since the original DDG method has been introduced by Liu et al.[8]in 2009,a variety of DDG type methods have been proposed and further developed.In this paper,we further investigate and develop a new DDG method with interface correction(DDG(IC))as the discretization of viscous and heat fluxes for the compressible Navier-Stokes equations on unstructured grids.Compared to the original DDG method,the newly developed DDG(IC)method demonstrates its superior in delivering the optimal order of accuracy under demanding situations.Strategies in extension and application of this newly developed DDG(IC)method for solving the compressible Navier-Stokes equations and special treatments designed for handling boundary viscous fluxes are presented and examined in this work.The performance of the new DDG method with interface correction is carefully evaluated and assessed through a number of typical test cases.Numerical experiments show that the new DDG method with interface correction can achieve the optimal order of accuracy on both uniform structured grids and nonuniform unstructured grids,which clearly indicates its potential for further applications of real engineering practices.

Solution Remapping Method with Lower Bound Preservation for Navier-Stokes Equations in Aerodynamic Shape Optimization

It is found that the solution remapping technique proposed in[Numer.Math.Theor.Meth.Appl.,2020,13(4)]and[J.Sci.Comput.,2021,87(3):1-26]does not work out for the Navier-Stokes equations with a high Reynolds number.The shape deformations usually reach several boundary layer mesh sizes for viscous flow,which far exceed one-layer mesh that the original method can tolerate.The direct application to Navier-Stokes equations can result in the unphysical pressures in remapped solutions,even though the conservative variables are within the reasonable range.In this work,a new solution remapping technique with lower bound preservation is proposed to construct initial values for the new shapes,and the global minimum density and pressure of the current shape which serve as lower bounds of the corresponding variables are used to constrain the remapped solutions.The solution distribution provided by the present method is proven to be acceptable as an initial value for the new shape.Several numerical experiments show that the present technique can substantially accelerate the flow convergence for large deformation problemswith 70%-80%CPU time reduction in the viscous airfoil drag minimization.