Solving Two-Mode Shallow Water Equations Using Finite Volume Methods

In this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407–432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches–two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme– and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.     

A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes

We propose a decoupled and positivity-preserving discrete duality finite volume (DDFV) scheme for anisotropic diffusion problems on polyhedral meshes with star-shaped cells and planar faces. Under the generalized DDFV framework, two sets of finite volume (FV) equations are respectively constructed on the dual and primary meshes, where the ones on the dual mesh are derived from the ingenious combination of a geometric relationship with the construction of the cell matrix. The resulting system on the dual mesh is symmetric and positive definite, while the one on the primary mesh possesses an M-matrix structure. To guarantee the positivity of the two categories of unknowns, a cutoff technique is introduced. As for the local conservation, it is conditionally maintained on the dual mesh while strictly preserved on the primary mesh. More interesting is that the FV equations on the dual mesh can be solved independently, so that the two sets of FV equations are decoupled. As a result, no nonlinear iteration is required for linear problems and a general nonlinear solver could be used for nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined by numerical experiments. The efficiency of the Newton method is also demonstrated by comparison with those of the fixed-point iteration method and its Anderson acceleration.     

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.     

On Invariant-Preserving Finite Difference Schemes for the Camassa-Holm Equation and the Two-Component Camassa-Holm System

The purpose of this paper is to develop and test novel invariant-preserving finite difference schemes for both the Camassa-Holm (CH) equation and one of its 2-component generalizations (2CH). The considered PDEs are strongly nonlinear, admitting soliton-like peakon solutions which are characterized by a slope discontinuity at the peak in the wave shape, and therefore suitable for modeling both short wave breaking and long wave propagation phenomena. The proposed numerical schemes are shown to preserve two invariants, momentum and energy, hence numerically producing wave solutions with smaller phase error over a long time period than those generated by other conventional methods. We first apply the scheme to the CH equation and showcase the merits of considering such a scheme under a wide class of initial data. We then generalize this scheme to the 2CH equation and test this scheme under several types of initial data.     

Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients

Extrapolation cascadic multigrid (EXCMG) method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linear finite element discretization. However, it is not trivial to generalize the vertex-centred EXCMG method to cell-centered finite volume (FV) methods for diffusion equations with strongly discontinuous and anisotropic coefficients, since a non-nested hierarchy of grid nodes are used in the cell-centered discretization. For cell-centered FV schemes, the vertex values (auxiliary unknowns) need to be approximated by cell-centered ones (primary unknowns). One of the novelties is to propose a new gradient transfer (GT) method of interpolating vertex unknowns with cell-centered ones, which is easy to implement and applicable to general diffusion tensors. The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method, and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients. Numerical experiments are presented to demonstrate the high efficiency of the proposed method.     

A Class of Hybrid DG/FV Methods for Conservation Laws III: Two-Dimensional Euler Equations

A concept of "static reconstruction" and "dynamic reconstruction" was introduced for higher-order (third-order or more) numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods had been developed for one-dimensional conservation law using a "hybrid reconstruction" approach, and extended to two-dimensional scalar equations on triangular and Cartesian/triangular hybrid grids. In the hybrid DG/FV schemes, the lower-order derivatives of the piecewise polynomial are computed locally in a cell by the traditional DG method (called as "dynamic reconstruction"), while the higher-order derivatives are reconstructed by the "static reconstruction" of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent neighboring cells. In this paper, the hybrid DG/FV schemes are extended to two-dimensional Euler equations on triangular and Cartesian/triangular hybrid grids. Some typical test cases are presented to demonstrate the performance of the hybrid DG/FV methods, including the standard vortex evolution problem with exact solution, isentropic vortex/weak shock wave interaction, subsonic flows past a circular cylinder and a three-element airfoil (30P30N), transonic flow past a NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV method achieves the desired third-order accuracy, and the applications demonstrate that they can capture the flow structure accurately, and can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy.     

An Efficient,Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step size. Owing to energy stability, we show that the scheme is stable in the time and space discrete$ℓ^∞$(0,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part – for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows.     

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique not only reduces the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.     

Numerical Study of Singularity Formation in Relativistic Euler Flows

The formation of singularities in relativistic flows is not well understood. Smooth solutions to the relativistic Euler equations are known to have a finite lifespan; the possible breakdown mechanisms are shock formation, violation of the subluminal conditions and mass concentration. We propose a new hybrid Glimm/central-upwind scheme for relativistic flows. The scheme is used to numerically investigate, for a family of problems, which of the above mechanisms is involved.     

Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation

In this paper, we propose a new conservative semi-Lagrangian (SL) finite difference (FD) WENO scheme for linear advection equations, which can serve as a base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume (FV) WENO scheme [3]. However, instead of inputting cell averages and approximate the integral form of the equation in a FV scheme, we input point values and approximate the differential form of equation in a FD spirit, yet retaining very high order (fifth order in our experiment) spatial accuracy. The advantage of using point values, rather than cell averages, is to avoid the second order spatial error, due to the shearing in velocity (v) and electrical field (E) over a cell when performing the Strang splitting to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy, compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear advection, rigid body rotation problem; and on the Landau damping and two-stream instabilities by solving the VP system. For comparison, we also apply (1) the conservative SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2) the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative version of the SL FD WENO scheme in [3] to the same test problems. The performances of different schemes are compared by the error table, solution resolution of sharp interface, and by tracking the conservation of physical norms, energies and entropies, which should be physically preserved.     

An Implicit Algorithm for High-Order DG/FV Schemes for Compressible Flows on 2D Arbitrary Grids

A Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit method was developed to solve two-dimensional Euler and Navier-Stokes equations by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed to solve the nonlinear system, while the linear system was solved with LU-SGS iteration. The effect of several parameters in the implicit scheme, such as the CFL number, the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated to evaluate the performance of convergence history. Several typical test cases were simulated, and compared with the traditional explicit Runge-Kutta (RK) scheme. Firstly the Couette flow was tested to validate the order of accuracy of the present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel was simulated and the effect of parameters was alsoinvestigated. Finally, the implicit algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder and the viscous flow in a square cavity. The numerical results demonstrated that the present implicit scheme can accelerate the convergence history efficiently. Choosing proper parameters would improve the efficiency of the implicit scheme. Moreover, in the same framework, the DG/FV hybrid schemes are more efficient than the same order DG schemes.     

A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms

In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non-stiff source terms as well as relaxation systems. We call the scheme a slope propagation (SP) method. It is an extension of our scheme derived for homogeneous hyperbolic systems [1]. In the present inhomogeneous systems the relaxation time may vary from order of one to a very small value. These small values make the relaxation term stronger and highly stiff. In such situations underresolved numerical schemes may produce spurious numerical results. However, our present scheme has the capability to correctly capture the behavior of the physical phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved. The scheme treats the space and time in a unified manner. The flow variables and their slopes are the basic unknowns in the scheme. The source term is treated by its volumetric integration over the space-time control volume and is a direct part of the overall space-time flux balance. We use two approaches for the slope calculations of the flow variables, the first one results directly from the flux balance over the control volumes, while in the second one we use a finite difference approach. The main features of the scheme are its simplicity, its Jacobian-free and Riemann solver-free recipe, as well as its efficiency and high order accuracy. In particular we show that the scheme has a discrete analog of the continuous asymptotic limit. We have implemented our scheme for various test models available in the literature such as the Broadwell model, the extended thermodynamics equations, the shallow water equations, traffic flow and the Euler equations with heat transfer. The numerical results validate the accuracy, versatility and robustness of the present scheme.     

A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space

In this paper, we develop a conservative numerical method for the Cahn– Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space. We use an unconditionally gradient stable nonlinear splitting numerical scheme and solve the resulting system of implicit discrete equations on a discrete narrow band domain by using a Jacobi-type iteration. For the domain boundary cells, we use the trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass. We perform numerical experiments on the various curved surfaces such as sphere, torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of the proposed method. We also present the dynamics of the CH equation with constant and space-dependent mobilities on the curved surfaces.     

A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro [25], in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical to the Godunov upwind method in applications to a 2×2 linear hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations. Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes. Finally, numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.     

Computational Aspects of Multiscale Simulation with the Lumped Particle Framework

First introduced in [2], the lumped particle framework is a flexible and numerically efficient framework for the modelling of particle transport in fluid flow. In this paper, the framework is expanded to simulate multicomponent particle-laden fluid flow. This is accomplished by introducing simulation protocols to model particles over a wide range of length and time scales. Consequently, we present a time ordering scheme and an approximate approach for accelerating the computation of evolution of different particle constituents with large differences in physical scales. We apply the extended framework on the temporal evolution of three particle constituents in sand-laden flow, and horizontal release of spherical particles. Furthermore, we evaluate the numerical error of the lumped particle model. In this context, we discuss the Velocity-Verlet numerical scheme, and show how to apply this to solving Newton's equations within the framework. We show that the increased accuracy of the Velocity-Verlet scheme is not lost when applied to the lumped particle framework.     

Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations

We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann (PB) equation as the equilibrium, in the external unbounded domain. To justify the simulation in a truncated domain, an error estimate of the truncation is proved in the symmetric cases for the PB equation. Then, the random batch interacting particle methods are introduced which are $\mathcal{O}(N)$ per time step. The particle methods can not only be considered as a numerical method for solving the PNP and PB equations, but also can be used as a direct simulation approach for the dynamics of the charged particles in solution. The particle methods are preferable due to their simplicity and adaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effects and interactions in the particle methods and to describe phenomena beyond the scope of the mean-field equations.     

Consistent Forcing Scheme in the Simplified Lattice Boltzmann Method for Incompressible Flows

Considering the fact that the lattice discrete effects are neglected while introducing a body force into the simplified lattice Boltzmann method (SLBM), we propose a consistent forcing scheme in SLBM for incompressible flows with external forces. The lattice discrete effects are considered at the level of distribution functions in the present forcing scheme. Consequently, it is more accurate compared with the original forcing scheme used in SLBM. Through Taylor series expansion and Chapman-Enskog (CE) expansion analysis, the present forcing scheme can be proven to recover the macroscopic Navier-Stokes (N-S) equations. Then, the macroscopic equations are resolved through a fractional step technique. Furthermore, the material derivative term is discretized by the central difference method. To verify the results of the present scheme, we simulate with multiple forms of external force interactions including the space- and time-dependent body forces. Hence, the present forcing scheme overcomes the disadvantages of the original forcing scheme and the body force can be accurately imposed in the present scheme even when a coarse mesh is applied while the original scheme fails. Excellent agreements between the analytical solutions and our numerical results can be observed.     

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.     

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.