A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.     

A Preliminary Calculation of Three-Dimensional Unsteady Underwater Cavitating Flows Near Incompressible Limit

Recently, cavitated flows over underwater submerged bodies have attracted researchers to simulate large scale cavitation. Comparatively Computational Fluid Dynamics (CFD) approaches have been used widely and successfully to model developed cavitation. However, it is still a great challenge to accurately predict cavitated flow phenomena associated with interface capturing, viscous effects, unsteadiness and three-dimensionality. In this study, we consider the preconditioned three-dimensional multiphase Navier-Stokes equations comprised of the mixture density, mixture momentum and constituent volume fraction equations. A dual-time implicit formulation with LU Decomposition is employed to accommodate the inherently unsteady physics. Also, we adopt the Roe flux splitting method to deal with flux discretization in space. Moreover, time-derivative preconditioning is used to ensure well-conditioned eigenvalues of the high density ratio two-phase flow system to achieve computational efficiency. Validation cases include an unsteady 3-D cylindrical headform cavitated flow and an 2-D convergent-divergent nozzle channel cavity-problem.     

On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations

Implicit time integration schemes are popular because their relaxed stability constraints can result in better computational efficiency. For time-accurate unsteady simulations, it has been well recognized that the inherent dispersion and dissipation errors of implicit Runge-Kutta schemes will reduce the computational accuracy for large time steps. Yet for steady state simulations using the time-dependent governing equations, these errors are often overlooked because the intermediate solutions are of less interest. Based on the model equation dy/dt = (µ+iλ)y of scalar convection diffusion systems, this study examines the stability limits, dispersion and dissipation errors of four diagonally implicit Runge-Kutta-type schemes on the complex (µ+iλ)∆t plane. Through numerical experiments, it is shown that, as the time steps increase, the A-stable implicit schemes may not always have reduced CPU time and the computations may not always remain stable, due to the inherent dispersion and dissipation errors of the implicit Runge-Kutta schemes. The dissipation errors may decelerate the convergence rate, and the dispersion errors may cause large oscillations of the numerical solutions. These errors, especially those of high wavenumber components, grow at large time steps. They lead to difficulty in the convergence of the numerical computations, and result in increasing CPU time or even unstable computations as the time step increases. It is concluded that an optimal implicit time integration scheme for steady state simulations should have high dissipation and low dispersion.     

Implicit Quadrature-Free Direct Reconstruction Method for Efficient Scale-Resolving Simulations

The present study develops implicit physical domain-based discontinuous Galerkin (DG) methods for efficient scale-resolving simulations on mixed-curved meshes. Implicit methods are essential to handle stiff systems in many scale-resolving simulations of interests in computational science and engineering. The physical domain-based DG method can achieve high-order accuracy using the optimal bases set and preserve the required accuracy on non-affine meshes. When using the quadrature-based DG method, these advantages are overshadowed by severe computational costs on mixed-curved meshes, making implicit scale-resolving simulations unaffordable. To address this issue, the quadrature-free direct reconstruction method (DRM) is extended to the implicit DG method. In this approach, the generalized reconstruction approximates non-linear flux functions directly in the physical domain, making the computing-intensive numerical integrations precomputable at a preprocessing step. The DRM operator is applied to the residual computation in the matrix-free method. The DRM operator can be further extended to the system matrix computation for the matrix-explicit Krylov subspace method and preconditioning. Finally, the A-stable Rosenbrock-type Runge–Kutta methods are adopted to achieve high-order accuracy in time. Extensive verification and validation from the manufactured solution to implicit large eddy simulations are conducted. The computed results confirm that the proposed method significantly improves computational efficiency compared to the quadrature-based method while accurately resolving detailed unsteady flow features that are hardly captured by scale-modeled simulations.     

A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh

High-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme's order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.     

The Mass-Preserving S-DDM Scheme for Two-Dimensional Parabolic Equations

In the paper, we develop and analyze a new mass-preserving splitting domain decomposition method over multiple sub-domains for solving parabolic equations. The domain is divided into non-overlapping multi-bock sub-domains. On the interfaces of sub-domains, the interface fluxes are computed by the semi-implicit (explicit) flux scheme. The solutions and fluxes in the interiors of sub-domains are computed by the splitting one-dimensional implicit solution-flux coupled scheme. The important feature is that the proposed scheme is mass conservative over multiple non-overlapping sub-domains. Analyzing the mass-preserving S-DDM scheme is difficult over non-overlapping multi-block sub-domains due to the combination of the splitting technique and the domain decomposition at each time step. We prove theoretically that our scheme satisfies conservation of mass over multi-block non-overlapping sub-domains and it is unconditionally stable. We further prove the convergence and obtain the error estimate in $L^2$-norm. Numerical experiments confirm theoretical results.     

Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation

The development of high-order schemes has been mostly concentrated on the limiters and high-order reconstruction techniques. In this paper, the effect of the flux functions on the performance of high-order schemes will be studied. Based on the same WENO reconstruction, two schemes with different flux functions, i.e., the fifth-order WENO method and the WENO-Gas-Kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variable reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volume WENO-GKS is a conservative variable reconstruction based method with the same WENO reconstruction. But it evaluates a time dependent gas distribution function along a cell interface, and updates the flow variables inside each control volume by integrating the flux function along the boundary of the control volume in both space and time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3 step problem, and the cavity flow at Reynolds number 3200. Our study shows that both WENO-SW and WENO-GKS yield quantitatively similar results and agree with each other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation and present more accurate solutions than those from the WENO-SW in all test cases. For the Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous terms in a single flux evaluation, and the WENO-SW uses an operator splitting technique, it appears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 times slower than the finite difference WENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore, it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-order gas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck for the further development of high-order schemes. In order to avoid the weakness of the low order flux function, the development of high-order schemes relies heavily on the weak solution of the original governing equations for the update of additional degree of freedom, such as the non-conservative gradients of flow variables, which cannot be physically valid in discontinuous regions.     

Grad-div Stabilized Finite Element Schemes for the Fluid-Fluid Interaction Model

In this work, two fully discrete grad-div stabilized finite element schemes for the fluid-fluid interaction model are considered. The first scheme is standard grad-div stabilized scheme, and the other one is modular grad-div stabilized scheme which adds to Euler backward scheme an update step and does not increase computational time for increasing stabilized parameters. Moreover, stability and error estimates of these schemes are given. Finally, computational tests are provided to verify both the numerical theory and efficiency of the presented schemes.     

Implementation of Finite Difference Weighted Compact Nonlinear Schemes with the Two-Stage Fourth-Order Accurate Temporal Discretization

In this paper, we present a new two-stage fourth-order finite difference weighted compact nonlinear scheme (WCNS) for hyperbolic conservation laws with special application to compressible Euler equations. To construct this algorithm, apart from the traditional WCNS for the spatial derivative, it was necessary to first construct a linear compact/explicit scheme utilizing time derivative of flux at midpoints, which, in turn, was solved by a generalized Riemann solver. Combining these two schemes, the fourth-order time accuracy was achieved using only the two-stage time-stepping technique. The final algorithm was numerically tested for various one-dimensional and two-dimensional cases. The results demonstrated that the proposed algorithm had an essentially similar performance as that based on the fourth-order Runge-Kutta method, while it required 25 percent less computational cost for one-dimensional cases, which is expected to decline further for multidimensional cases.     

Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible fluids. In order to obtain a sufficiently stable higher order scheme with respect to the time and space coordinates, we develop a combination of the discontinuous Galerkin finite element (DGFE) method for the space discretization and the backward difference formulae (BDF) for the time discretization. Since the resulting discrete problem leads to a system of nonlinear algebraic equations at each time step, we employ suitable linearizations of inviscid as well as viscous fluxes which give a linear algebraic problem at each time step. Finally, the resulting BDF-DGFE scheme is applied to steady as well as unsteady flows and achieved results are compared with reference data.     

A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows

In this paper, a new sharp-interface approach to simulate compressible multiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservative discontinuous Galerkin finite element scheme to evolve an indicator function that tracks the material interface. At the interface our method applies ghost cells to compute the numerical flux, as the ghost fluid method. However, unlike the original ghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derived from an approximate-state Riemann solver, similar to the approach proposed in [25], but based on a much simpler formulation. Our formulation leads only to one single scalar nonlinear algebraic equation that has to be solved at the interface, instead of the system used in [25]. Away from the interface, we use the new general Osher-type flux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemann solver, applicable to general hyperbolic conservation laws. The time integration is performed using a fully-discrete one-step scheme, based on the approaches recently proposed in [5, 7]. This allows us to evolve the system also with time-accurate local time stepping. Due to the sub-cell resolution and the subsequent more restrictive time-step constraint of the DG scheme, a local evolution for the indicator function is applied, which is matched with the finite volume scheme for the solution of the Euler equations that runs with a larger time step. The use of a locally optimal time step avoids the introduction of excessive numerical diffusion in the finite volume scheme. Two different fluids have been used, namely an ideal gas and a weakly compressible fluid modeled by the Tait equation. Several tests have been computed to assess the accuracy and the performance of the new high order scheme. A verification of our algorithm has been carefully carried out using exact solutions as well as a comparison with other numerical reference solutions. The material interface is resolved sharply and accurately without spurious oscillations in the pressure field.     

Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations

In this paper, we will develop a fast iterative solver for the system of linear equations arising from the local discontinuous Galerkin (LDG) spatial discretization and additive Runge-Kutta (ARK) time marching method for the KdV type equations. Being implicit in time, the severe time step ($∆t$=$\mathcal{O}(∆x^k)$, with the $k$-th order of the partial differential equations (PDEs)) restriction for explicit methods will be removed. The equations at the implicit time level are linear and we demonstrate an efficient, practical multigrid (MG) method for solving the equations. In particular, we numerically show the optimal or sub-optimal complexity of the MG solver and a two-level local mode analysis is used to analyze the convergence behavior of the MG method. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency and capability of the LDG method coupled with the multigrid method for solving the KdV type equations.     

Operator Splitting for Three-Phase Flow in Heterogeneous Porous Media

We describe an operator splitting technique based on physics rather than on dimension for the numerical solution of a nonlinear system of partial differential equations which models three-phase flow through heterogeneous porous media. The model for three-phase flow considered in this work takes into account capillary forces, general relations for the relative permeability functions and variable porosity and permeability fields. In our numerical procedure a high resolution, nonoscillatory, second order, conservative central difference scheme is used for the approximation of the nonlinear system of hyperbolic conservation laws modeling the convective transport of the fluid phases. This scheme is combined with locally conservative mixed finite elements for the numerical solution of the parabolic and elliptic problems associated with the diffusive transport of fluid phases and the pressure-velocity problem. This numerical procedure has been used to investigate the existence and stability of nonclassical shock waves (called transitional or undercompressive shock waves) in two-dimensional heterogeneous flows, thereby extending previous results for one-dimensional flow problems. Numerical experiments indicate that the operator splitting technique discussed here leads to computational efficiency and accurate numerical results.     

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In addition, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.     

A Multigrid Method for a Model of the Implicit Immersed Boundary Equations

Explicit time stepping schemes for the immersed boundary method require very small time steps in order to maintain stability. Solving the equations that arise from an implicit discretization is difficult. Recently, several different approaches have been proposed, but a complete understanding of this problem is still emerging. A multigrid method is developed and explored for solving the equations in an implicit-time discretization of a model of the immersed boundary equations. The model problem consists of a scalar Poisson equation with conformation-dependent singular forces on an immersed boundary. This model does not include the inertial terms or the incompressibility constraint. The method is more efficient than an explicit method, but the efficiency gain is limited. The multigrid method alone may not be an effective solver, but when used as a preconditioner for Krylov methods, the speed-up over the explicit-time method is substantial. For example, depending on the constitutive law for the boundary force, with a time step 100 times larger than the explicit method, the implicit method is about 15-100 times more efficient than the explicit method. A very attractive feature of this method is that the efficiency of the multigrid preconditioned Krylov solver is shown to be independent of the number of immersed boundary points.     

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree $N$ inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.     

IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows

We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.     

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows III: Microflow Simulations

Due to the rapid advances in micro-electro-mechanical systems (MEMS), the study of microflows becomes increasingly important. Currently, the molecular-based simulation techniques are the most reliable methods for rarefied flow computation, even though these methods face statistical scattering problem in the low speed limit. With discretized particle velocity space, a unified gas-kinetic scheme (UGKS) for entire Knudsen number flow has been constructed recently for flow computation. Contrary to the particle-based direct simulation Monte Carlo (DSMC) method, the unified scheme is a partial differential equation-based modeling method, where the statistical noise is totally removed. But the common point between the DSMC and UGKS is that both methods are constructed through direct modeling in the discretized space. Due to the multiscale modeling in the unified method, i.e., the update of both macroscopic flow variables and microscopic gas distribution function, the conventional constraint of time step being less than the particle collision time in many direct Boltzmann solvers is released here. The numerical tests show that the unified scheme is more efficient than the particle-based methods in the low speed rarefied flow computation. The main purpose of the current study is to validate the accuracy of the unified scheme in the capturing of non-equilibrium flow phenomena. In the continuum and free molecular limits, the gas distribution function used in the unified scheme for the flux evaluation at a cell interface goes to the corresponding Navier-Stokes and free molecular solutions. In the transition regime, the DSMC solution will be used for the validation of UGKS results. This study shows that the unified scheme is indeed a reliable and accurate flow solver for low speed non-equilibrium flows. It not only recovers the DSMC results whenever available, but also provides high resolution results in cases where the DSMC can hardly afford the computational cost. In thermal creep flow simulation, surprising solution, such as the gas flowing from hot to cold regions along the wallsurface, is observed for the first time by the unified scheme, which is confirmed later through intensive DSMC computation.     

A Scalable Numerical Method for Simulating Flows Around High-Speed Train Under Crosswind Conditions

This paper presents a parallel Newton-Krylov-Schwarz method for the numerical simulation of unsteady flows at high Reynolds number around a high-speed train under crosswind. With a realistic train geometry, a realistic Reynolds number, and a realistic wind speed, this is a very challenging computational problem. Because of the limited parallel scalability, commercial CFD software is not suitable for supercomputers with a large number of processors. We develop a Newton-Krylov-Schwarz based fully implicit method, and the corresponding parallel software, for the 3D unsteady incompressible Navier-Stokes equations discretized with a stabilized finite element method on very fine unstructured meshes. We test the algorithm and software for flows passing a train modeled after China's high-speed train CRH380B, and we also compare our results with results obtained from commercial CFD software. Our algorithm shows very good parallel scalability on a supercomputer with over one thousand processors.     

Adaptive Fully Implicit Simulator with Multilevel Schwarz Methods for Gas Reservoir Flows in Fractured Porous Media

Large-scale reservoir modeling and simulation of gas reservoir flows in fractured porous media is currently an important topic of interest in petroleum engineering. In this paper, the dual-porosity dual-permeability (DPDP) model coupled with the Peng-Robinson equation of state (PR-EoS) is used for the mathematical model of the gas reservoir flow in fractured porous media. We develop and study a parallel and highly scalable reservoir simulator based on an adaptive fully implicit scheme and an inexact Newton type method to solve this dual-continuum mathematical model. In the approach, an explicit-first-step, single-diagonal-coefficient, diagonally implicit Runge–Kutta (ESDIRK) method with adaptive time stepping is proposed for the fully implicit discretization, which is second-order and L-stable. And then we focus on the family of Newton–Krylov methods for the solution of a large sparse nonlinear system of equations arising at each time step. To accelerate the convergence and improve the scalability of the solver, a class of multilevel monolithic additive Schwarz methods is employed for preconditioning. Numerical results on a set of ideal as well as realistic flow problems are used to demonstrate the efficiency and the robustness of the proposed methods. Experiments on a supercomputer with several thousand processors are also carried out to show that the proposed reservoir simulator is highly scalable.