High-order discontinuous Galerkin method for applications to multicomponent and chemically reacting flows

This article focuses on the development of a discontinuous Galerkin (DG) method for simulations of multicomponent and chemically reacting flows. Compared to aerodynamic flow applications, in which DG methods have been successfully employed, DG simulations of chemically reacting flows introduce challenges that arise from flow unsteadiness, combustion, heat release, compressibility effects, shocks, and variations in thermodynamic properties. To address these challenges, algorithms are developed, including an entropy-bounded DG method, an entropy-residual shock indicator, and a new formulation of artificial viscosity. The performance and capabilities of the resulting DG method are demonstrated in several relevant applications, including shock/bubble interaction, turbulent combustion, and detonation. It is concluded that the developed DG method shows promising performance in application to multicomponent reacting flows. The paper concludes with a discussion of further research needs to enable the application of DG methods to more complex reacting flows.

High-order discontinuous Galerkin solver on hybrid anisotropic meshes for laminar and turbulent simulations

Efficient and robust solution strategies are developed for discontinuous Galerkin （DG） discretization of the Navier-Stokes （NS） and Reynolds-averaged NS （RANS） equations on structured/unstructured hybrid meshes. A novel line-implicit scheme is devised and implemented to reduce the memory gain and improve the computational eificiency for highly anisotropic meshes. A simple and effective technique to use the mod- ified Baldwin-Lomax （BL） model on the unstructured meshes for the DC methods is proposed. The compact Hermite weighted essentially non-oscillatory （HWENO） limiters are also investigated for the hybrid meshes to treat solution discontinuities. A variety of compressible viscous flows are performed to examine the capability of the present high- order DG solver. Numerical results indicate that the designed line-implicit algorithms exhibit weak dependence on the cell aspect-ratio as well as the discretization order. The accuracy and robustness of the proposed approaches are demonstrated by capturing com- plex flow structures and giving reliable predictions of benchmark turbulent problems.

3D NUMERICAL SOLUTION OF AERO-NOISE WITH HIGH-ORDER DISCONTINUOUS GALERKIN METHOD

The flow-induced noise is simulated with a hybrid method.Firstly,a steady-state background flow field is given by solving Reynolds averaged Navier-Stokes(RANS)equations with finite volume(FV)method on structured grid.Then the linearized Euler equations(LEE)can be constructed based on the resulted background flow field,where the source term on the right hand side is computed using stochastic noise generation and radiation(SNGR)method.Finally,the unsteady acoustic field is obtained through solving LEE using high-order discontinuous Galerkin(DG)method on unstructured grid,where the parallel computing based on mesh partitioning and a″Quadrature-Free Implementation″method for high-order DG are employed to accelerate the computation.In order to demonstrate the sound propagation in detail,a visualization method for high-order schemes is also developed here.Moreover,in order to test the validation and the accuracy,a 3D cavity test in comparison with the experimental data is displayed first in this paper,then a 3D high-lift wing is also simulated to demonstrate its capability for very complex geometries.

Implicit high-order discontinuous Galerkin method with HWENO type limiters for steady viscous flow simulations

Two types of implicit algorithms have been im- proved for high order discontinuous Galerkin （DG） method to solve compressible Navier-Stokes （NS） equations on tri- angular grids. A block lower-upper symmetric Gauss- Seidel （BLU-SGS） approach is implemented as a nonlin- ear iterative scheme. And a modified LU-SGS （LLU-SGS） approach is suggested to reduce the memory requirements while retain the good convergence performance of the origi- nal LU-SGS approach. Both implicit schemes have the sig- nificant advantage that only the diagonal block matrix is stored. The resulting implicit high-order DG methods are applied, in combination with Hermite weighted essentially non-oscillatory （HWENO） limiters, to solve viscous flow problems. Numerical results demonstrate that the present implicit methods are able to achieve significant efficiency improvements over explicit counterparts and for viscous flows with shocks, and the HWENO limiters can be used to achieve the desired essentially non-oscillatory shock tran- sition and the designed high-order accuracy simultaneously.

High-Order Discontinuous Galerkin Solution of Compressible Flows with a Hybrid Lattice Boltzmann Flux

A discontinuous Galerkin(DG)-based lattice Boltzmann method is employed to solve the Euler and Navier-Stokes equations.Instead of adopting the widely used local Lax-Friedrichs flux and Roe Flux etc.,a hybrid lattice Boltzmann flux solver(LBFS)is employed to evaluate the inviscid flux across the cell interfaces.The main advantage of the hybrid LBFS is its flexibility for capturing both strong shocks and thin boundary layers through introducing a function which varies from zero to one to control the artificial viscosity.Numerical results indicate that the hybrid lattice Boltzmann flux solver behaves very well combining with the high-order DG method when simulating both inviscid and viscous flows.

High-Order Discontinuous Galerkin Method for Hovering Rotor Simulations Based on a Rotating Reference Frame

An implicit higher ? order discontinuous Galerkin(DG) spatial discretization for the compressible Euler equations in a rotating frame of reference is presented and applied to a rotor in hover using hexahedral grids. Instead of auxiliary methods like grid adaptation,higher ? order simulations(fourth ? and fifth ? order accuracy) are adopted.Rigorous numerical experiments are carefully designed,conducted and analyzed. The results show generally excellent consistence with references and vigorously demonstrate the higher?order DG method's better performance in loading distribution computations and tip vortex capturing, with much fewer degrees of freedom(DoF). Detailed investigations on the outer boundary conditions for hovering rotors are presented as well. A simple but effective speed smooth procedure is developed specially for the DG method. Further results reveal that the rarely used pressure restriction for outlet speed has a considerable advantage over the extensively adopted vertical speed restriction.

A robust implicit high-order discontinuous Galerkin method for solving compressible Navier-Stokes equations on arbitrary grids

The primary impediments impeding the implementation of high-order methods in simulating viscous flow over complex configurations are robustness and convergence.These challenges impose significant constraints on computational efficiency,particularly in the domain of engineering applications.To address these concerns,this paper proposes a robust implicit high-order discontinuous Galerkin(DG)method for solving compressible Navier-Stokes(NS)equations on arbitrary grids.The method achieves a favorable equilibrium between computational stability and efficiency.To solve the linear system,an exact Jacobian matrix solving strategy is employed for preconditioning and matrix-vector generation in the generalized minimal residual(GMRES)method.This approach mitigates numerical errors in Jacobian solution during implicit calculations and facilitates the implementation of an adaptive Courant-Friedrichs-Lewy(CFL)number increasing strategy,with the aim of improving convergence and robustness.To further enhance the applicability of the proposed method for intricate grid distortions,all simulations are performed in the reference domain.This practice significantly improves the reversibility of the mass matrix in implicit calculations.A comprehensive analysis of various parameters influencing computational stability and efficiency is conducted,including CFL number,Krylov subspace size,and GMRES convergence criteria.The computed results from a series of numerical test cases demonstrate the promising results achieved by combining the DG method,GMRES solver,exact Jacobian matrix,adaptive CFL number,and reference domain calculations in terms of robustness,convergence,and accuracy.These analysis results can serve as a reference for implicit computation in high-order calculations.

High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.

A Straightforward hp-Adaptivity Strategy for Shock-Capturing with High-Order Discontinuous Galerkin Methods

In this paper,high-order Discontinuous Galerkin(DG)method is used to solve the two-dimensional Euler equations.A shock-capturing method based on the artificial viscosity technique is employed to handle physical discontinuities.Numerical tests show that the shocks can be captured within one element even on very coarse grids.The thickness of the shocks is dominated by the local mesh size and the local order of the basis functions.In order to obtain better shock resolution,a straightforward hp-adaptivity strategy is introduced,which is based on the high-order contribution calculated using hierarchical basis.Numerical results indicate that the hp-adaptivity method is easy to implement and better shock resolution can be obtained with smaller local mesh size and higher local order.

High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

A high-order leap-frog based non-dissipative discontinuous Galerkin time- domain method for solving Maxwell＇s equations is introduced and analyzed. The pro- posed method combines a centered approximation for the evaluation of fluxes at the in- terface between neighboring elements, with a Nth-order leap-frog time scheme. More- over, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwelrs equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with high- order elements show the potential of the method.

Capturing Near-Equilibrium Solutions: A Comparison between High-Order Discontinuous Galerkin Methods and Well-Balanced Schemes

Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-balanced approach. We present a modern numerical implementation of these two strategies and compare them in details, using hydrostatic but also dynamical equilibrium solutions of several simple test cases. Finally, we apply our methodology to the simulation of a protoplanetary disc in centrifugal equilibrium around its star and model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.

A High-Order Discontinuous Galerkin Solver for Helically Symmetric Flows

We present a high-order discontinuous Galerkin(DG)scheme to solve the system of helically symmetric Navier-Stokes equations which are discussed in[28].In particular,we discretize the helically reduced Navier-Stokes equations emerging from a reduction of the independent variables such that the remaining variables are:t,r,ξwithξ=az+bϕ,where r,ϕ,z are common cylindrical coordinates and t the time.Beside this,all three velocity components are kept non-zero.A new non-singular coordinateηis introduced which ensures that a mapping of helical solutions into the three-dimensional space is well defined.Using that,periodicity conditions for the helical frame aswell as uniqueness conditions at the centerline axis r=0 are derived.In the sector near the axis of the computational domain a change of the polynomial basis is implemented such that all physical quantities are uniquely defined at the centerline.For the temporal integration,we present a semi-explicit scheme of third order where the full spatial operator is splitted into a Stokes operator which is discretized implicitly and an operator for the nonlinear terms which is treated explicitly.Computations are conducted for a cylindrical shell,excluding the centerline axis,and for the full cylindrical domain,where the centerline is included.In all cases we obtain the convergence rates of order O(hk+1)that are expected from DG theory.In addition to the first DG discretization of the system of helically invariant Navier-Stokes equations,the treatment of the central axis,the resulting reduction of the DG space,and the simultaneous use of a semi-explicit time stepper are of particular novelty.

Performance Analysis of a High-Order Discontinuous Galerkin Method Application to the Reverse Time Migration

This work pertains to numerical aspects of a finite element method based discontinuous functions.Our study focuses on the Interior Penalty Discontinuous Galerkin method(IPDGM)because of its high-level of flexibility for solving the full wave equation in heterogeneousmedia.We assess the performance of IPDGMthrough a comparison study with a spectral element method(SEM).We show that IPDGM is as accurate as SEM.In addition,we illustrate the efficiency of IPDGM when employed in a seismic imaging process by considering two-dimensional problems involving the Reverse Time Migration.

Adaptive Sparse Grid Discontinuous Galerkin Method:Review and Software Implementation

This paper reviews the adaptive sparse grid discontinuous Galerkin(aSG-DG)method for computing high dimensional partial differential equations(PDEs)and its software implementation.The C++software package called AdaM-DG,implementing the aSG-DG method,is available on GitHub at https://github.com/JuntaoHuang/adaptive-multiresolution-DG.The package is capable of treating a large class of high dimensional linear and nonlinear PDEs.We review the essential components of the algorithm and the functionality of the software,including the multiwavelets used,assembling of bilinear operators,fast matrix-vector product for data with hierarchical structures.We further demonstrate the performance of the package by reporting the numerical error and the CPU cost for several benchmark tests,including linear transport equations,wave equations,and Hamilton-Jacobi(HJ)equations.

Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.

A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.

A Local Macroscopic Conservative(LoMaC)Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics

In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank tensor method with discontinuous Galerkin(DG)discretization for the physical and phase spaces for simulating the Vlasov-Poisson(VP)system.The LoMaC property refers to the exact local conservation of macroscopic mass,momentum,and energy at the discrete level.The recently developed LoMaC low rank tensor algorithm(arXiv:2207.00518)simultaneously evolves the macroscopic conservation laws of mass,momentum,and energy using the kinetic flux vector splitting;then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables.This paper is a generalization of our previous work,but with DG discretization to take advantage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term.The algorithm is developed in a similar fashion as that for a finite difference scheme,by observing that the DG method can be viewed equivalently in a nodal fashion.With the nodal DG method,assuming a tensorized computational grid,one will be able to(i)derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms,and(ii)define a weighted inner product space based on the nodal DG grid points.The algorithm can be extended to the high dimensional problems by hierarchical Tucker(HT)decomposition of solution tensors and a corresponding conservative projection algorithm.In a similar spirit,the algorithm can be extended to DG methods on nodal points of an unstructured mesh,or to other types of discretization,e.g.,the spectral method in velocity direction.Extensive numerical results are performed to showcase the efficacy of the method.

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.

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.

Discontinuous Galerkin Method for Hydrodynamic and Sediment Transport Model

In this article,we propose and research a first-order,linearized discontinuous Galerkin method for the approximation of the hydrodynamic and sediment transport model.The method is decoupled and fully discrete,and is shown to be unconditionally stable.Furthermore,error estimates are proved.Finally,the theoretical analysis is confirmed by numerical examples.