The two-level element free Galerkin method for MHD flow at high Hartmann numbers

In this Letter, a new element free Galerkin method, namely the two-level element free Galerkin method, is presented for solving the governing equations of steady magnetohydrodynamic duct flow. Because this element free Galerkin method makes use of the nodal point configurations which do not require a mesh, therefore it differs from FEM-like approaches by avoiding the need of meshing, a very demanding task for complicated geometry problems. Another distinguished feature of the proposed method is the resolving capability of high gradients near the layer regions without local or adaptive refinements. Numerical results indicate that no matter how large the Hartmann number is, this method has the ability to produce the satisfactory results for the velocity and the magnetic field simultaneously. That is to say, the presented method has some excellent properties, such as better stability and accuracy.     

A finite volume method for approximating 3D diffusion operators on general meshes

A finite volume method is presented for discretizing 3D diffusion operators with variable full tensor coefficients. This method handles anisotropic, non-symmetric or discontinuous variable tensor coefficients while distorted, non-matching or non-convex n-faced polyhedron meshes can be used. For meshes of polyhedra whose faces have not more than four edges, the associated matrix is positive definite (and symmetric if the diffusion tensor is symmetric). A second-order (resp. first-order) accuracy is numerically observed for the solution (resp. gradient of the solution).     

A conservative discrete compatibility-constraint low-Mach pressure-correction algorithm for time-accurate simulations of variable density flows

In this paper, we develop the discrete compatibility-constraint pressure-correction algorithm for transient simulations of variable density flows at low-Mach numbers. The constraint for the velocity field is constructed from a combination of the discrete equations of continuity and scalar (e.g. energy) transport, imposing that the newly predicted state must be compatible, in agreement with the equation of state. This way, mass and scalar conservation are guaranteed and the equation of state is exactly fulfilled at every time step. For comparison reasons, two other types of well-known pressure-correction algorithms are also used. The first class, denoted as continuity-constraint pressure-correction, is based on a constraint for the velocity field that is derived solely from the continuity equation. The second class, denoted as analytical compatibility-constraint pressure-correction, constructs the constraint from an analytical combination of the material derivative of the equation of state and the continuity and scalar equations. The algorithms are tested for three example fluid configurations: a single-fluid ideal gas, a two-fluid inert mixture and a two-fluid reacting mixture. The latter is special in the sense that the equation of state is non-linear and not everywhere differentiable. The continuity-constraint pressure-correction algorithm yields unstable solutions if density ratios are high. The analytical compatibility-constraint pressure-correction algorithm yields stable results, but the predicted states do not correspond to the equation of state. The discrete compatibility-constraint pressure-correction algorithm performs well on all test cases: the simulation results are stable and exactly match the equation of state.     

The finite element method for the coupled Schrödinger-KdV equations

In this Letter, we employ finite element method to study a periodic initial value problem for the coupled Schrödinger-KdV equations. For the case of one dimension, this problem is reduced to a system of ordinary differential equations by using a semi-discrete scheme. The conservation properties of this scheme, the existence and uniqueness of the discrete solutions, and error estimates are presented. In numerical experiments, the resulting system of ordinary differential equations are solved by Runge-Kutta method at each time level. The superior accuracy of this scheme is shown by comparing the numerical solutions with the exact solutions.     

A high-order low-dispersion symmetry-preserving finite-volume method for compressible flow on curvilinear grids

A new high-order finite-volume method is presented that preserves the skew symmetry of convection for the compressible flow equations. The method is intended for Large-Eddy Simulations (LES) of compressible turbulent flows, in particular in the context of hybrid RANS–LES computations. The method is fourth-order accurate and has low numerical dissipation and dispersion. Due to the finite-volume approach, mass, momentum, and total energy are locally conserved. Furthermore, the skew-symmetry preservation implies that kinetic energy, sound-velocity, and internal energy are all locally conserved by convection as well. The method is unique in that all these properties hold on non-uniform, curvilinear, structured grids. Due to the conservation of kinetic energy, there is no spurious production or dissipation of kinetic energy stemming from the discretization of convection. This enhances the numerical stability and reduces the possible interference of numerical errors with the subgrid-scale model. By minimizing the numerical dispersion, the numerical errors are reduced by an order of magnitude compared to a standard fourth-order finite-volume method.     

The finite volume local evolution Galerkin method for solving the hyperbolic conservation laws

This paper presents a finite volume local evolution Galerkin (FVLEG) scheme for solving the hyperbolic conservation laws. The FVLEG scheme is the simplification of the finite volume evolution Galerkin method (FVEG). In FVEG, a necessary step is to compute the dependent variables at cell interfaces at t_n + τ (0 < τ ? Δt). The FVLEG scheme is constructed by taking τ → 0 in the evolution operators of FVEG. The FVLEG scheme greatly simplifies the evaluation of the numerical fluxes. It is also well suited with the semi-discrete finite volume method, making the flux evaluation being decoupled with the reconstruction procedure while maintaining the genuine multi-dimensional nature of the FVEG methods. The derivation of the FVLEG scheme is presented in detail. The performance of the proposed scheme is studied by solving several test cases. It is shown that FVLEG scheme can obtain very satisfactory numerical results in terms of accuracy and resolution.     

Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

A domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension, based on the probabilistic representation of solutions as an average of suitable multiplicative functionals. Such a direct probabilistic representation requires generating a number of random trees, whose role is that of the realizations of stochastic processes used in the linear problems. First, only few values of the sought solution inside the space-time domain are computed (by a Monte Carlo method on the trees). An interpolation is then carried out, in order to approximate interfacial values of the solution inside the domain. Thus, a fully decoupled set of sub-problems is obtained. The algorithm is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Pruning the trees is shown to increase appreciably the efficiency of the algorithm. Numerical examples conducted in 2D, including some for the KPP equation, are given.     

A roe-average algorithm for a granular-gas model with non-conservative terms

A Roe-average algorithm has been derived for a granular-gas model, proposed by Goldshtein and Shapiro [Goldshtein, Shapiro, Mechanics of collisional motion of granular materials: Part 1. General hydrodynamic equations, J. Fluid Mech. 282 (1995) 75–114], which contains non-conservative terms in the Euler-like hyperbolic governing equations apart from sink terms, which arise from inelastic collision of granules and are present only in the energy equation. The non-conservative terms introduce non-isentropic effects in acoustic-wave propagation within granular media and they also contribute to the Rankine–Hugoniot relations across a discontinuity. A Roe-average algorithm, based on the same granular-gas model, was derived in the literature [V. Kamenetsky, A. Goldshtein, M. Shapiro, D. Degani, Evolution of a shock wave in a granular gas, Phys. Fluids, 12 (2000) 3036–3049] which then required the implementation of a shock-fitting technique at a discontinuity. In the present work, Roe-averaged variables have been obtained from the Rankine–Hugoniot jump relations and the non-conservative terms have been incorporated in the numerical flux formula consistent with upwind principles associated with the granular speed of sound. Results for unsteady one-dimensional granular flows, colliding with a wall, demonstrate the capability of the proposed algorithm to capture strong shocks in addition to flow features not found in molecular gases, such as a fluidized region downstream of the shock and a compacted solid-block region adjacent to the wall.     

On exterior boundary-value problems of the Helmholtz equation not deduced from waves

A boundary integral equation is applied to describe a special kind of exterior Helmholtz boundary-value problem that is not deduced from waves. Then the asymptotic property of O(r ^–2) decay at infinity and the uniqueness of the solution as well as its finite energy property are discussed.     

An accurate adaptive solver for surface-tension-driven interfacial flows

A method combining an adaptive quad/octree spatial discretisation, geometrical Volume-Of-Fluid interface representation, balanced-force continuum-surface-force surface-tension formulation and height-function curvature estimation is presented. The extension of these methods to the quad/octree discretisation allows adaptive variable resolution along the interface and is described in detail. The method is shown to recover exact equilibrium (to machine accuracy) between surface-tension and pressure gradient in the case of a stationary droplet, irrespective of viscosity and spatial resolution. Accurate solutions are obtained for the classical test case of capillary wave oscillations. An application to the capillary breakup of a jet of water in air further illustrates the accuracy and efficiency of the method. The source code of the implementation is freely available as part of the Gerris flow solver.     

A finite element variational multiscale method for incompressible flows based on two local gauss integrations

In this article, we present a finite element variational multiscale (VMS) method for incompressible flows based on two local Gauss integrations, and compare it with common VMS method which is defined by a low order finite element space L_h$L_h$ on the same grid as X_h$X_h$ for the velocity deformation tensor and a stabilization parameter α$\alpha$. The best algorithmic feature of our method is using two local Gauss integrations to replace projection operator. We theoretically discuss the relationship between our method and common VMS method for the Taylor–Hood elements, and show that the nonlinear system derived from our method by finite element discretization is much smaller than that of common VMS method computationally.     

A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

The goal of this paper is to present high-order cell-centered schemes for solving the equations of Lagrangian gas dynamics written in cylindrical geometry. A node-based discretization of the numerical fluxes is obtained through the computation of the time rate of change of the cell volume. It allows to derive finite volume numerical schemes that are compatible with the geometric conservation law (GCL). Two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume scheme while the second one corresponds to the so-called area-weighted scheme. Both formulations share the same discretization for the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The control volume scheme is conservative for momentum, total energy and satisfies a local entropy inequality in its first-order semi-discrete form. However, it does not preserve spherical symmetry. On the other hand, the area-weighted scheme is conservative for total energy and preserves spherical symmetry for one-dimensional spherical flow on equi-angular polar grid. The two-dimensional high-order extensions of these two schemes are constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess these new schemes. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new schemes.     

A high-order super-grid-scale absorbing layer and its application to linear hyperbolic systems

We continue the development of the super-grid-scale model initiated in [T. Colonius, H. Ran, A super-grid-scale model for simulating compressible flow on unbounded domains, J. Comput. Phys. 182 (1) (2002) 191–212] and consider its application to linear hyperbolic systems. The super-grid-scale model consists of two parts: reduction of an unbounded to a bounded domain by a smooth coordinate transformation and a damping of those scales. For linear problems the super-grid scales are analogous to spurious numerical waves. We damp these waves by high-order undivided differences. We compute reflection coefficients for different orders of the damping and find that significant improvements are obtained when high-order damping is used.     

Compact finite difference method for the fractional diffusion equation

High-order compact finite difference scheme for solving one-dimensional fractional diffusion equation is considered in this paper. After approximating the second-order derivative with respect to space by the compact finite difference, we use the Grünwald–Letnikov discretization of the Riemann–Liouville derivative to obtain a fully discrete implicit scheme. We analyze the local truncation error and discuss the stability using the Fourier method, then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.     

A high order kinetic flux-vector splitting method for the reduced five-equation model of compressible two-fluid flows

We present a high order kinetic flux-vector splitting (KFVS) scheme for the numerical solution of a conservative interface-capturing five-equation model of compressible two-fluid flows. This model was initially introduced by Wackers and Koren (2004) [21]. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term in order to account for the energy exchange. We numerically investigate both one- and two-dimensional flow models. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the system of equations. In two space dimensions the scheme is derived in a usual dimensionally split manner. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge–Kutta time stepping method. For validation, the results of our scheme are compared with those from the high resolution central scheme of Nessyahu and Tadmor [14]. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential for modeling two-phase flows.     

A grid based particle method for evolution of open curves and surfaces

We propose a new numerical method for modeling motion of open curves in two dimensions and open surfaces in three dimensions. Following the grid based particle method we proposed in [S. Leung, H.K. Zhao, A grid based particle method for moving interface problems. J. Comput. Phys. 228 (2009) 2993–3024], we represent the open curve or the open surface by meshless Lagrangian particles sampled according to an underlying fixed Eulerian mesh. The underlying grid is used to provide a quasi-uniform sampling and neighboring information for meshless particles. The key idea in the current paper is to represent and to track end-points of the open curve and boundary-points of the open surface explicitly and consistently with interior particles. We apply our algorithms to several applications including spiral crystal growth modeling and image segmentation using active contours.     

A non-conforming monolithic finite element method for problems of coupled mechanics

In this study, a Lagrange multiplier technique is developed to solve problems of coupled mechanics and is applied to the case of a Newtonian fluid coupled to a quasi-static hyperelastic solid. Based on theoretical developments in [57], an additional Lagrange multiplier is used to weakly impose displacement/velocity continuity as well as equal, but opposite, force. Through this approach, both mesh conformity and kinematic variable interpolation may be selected independently within each mechanical body, allowing for the selection of grid size and interpolation most appropriate for the underlying physics. In addition, the transfer of mechanical energy in the coupled system is proven to be conserved. The fidelity of the technique for coupled fluid–solid mechanics is demonstrated through a series of numerical experiments which examine the construction of the Lagrange multiplier space, stability of the scheme, and show optimal convergence rates. The benefits of non-conformity in multi-physics problems is also highlighted. Finally, the method is applied to a simplified elliptical model of the cardiac left ventricle.     

A coupled edge-/face-based smoothed finite element method for structural-acoustic problems

The edge-based smoothed finite element method (ES-FEM) and the face-based smoothed finite element method (FS-FEM) developed recently have shown great efficiency in solving solid mechanics problems with triangular and tetrahedral meshes. In this paper, a coupled ES-/FS-FEM model is extended to solve the structural-acoustic problems consisting of a plate structure interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements are used to discretize the two-dimensional (2D) plate and three-dimensional (3D) fluid, respectively, as they can be generated easily and even automatically for complicated geometries. The field variable in each element is approximated using the linear shape functions, which is exactly the same as that in the standard FEM. The gradient field of the problem is obtained particularly using the gradient smoothing operation over the edge-based and face-based smoothing domains in 2D and 3D, respectively. The gradient smoothing technique can provide a proper softening effect to the model, effectively solve the problems caused by the well-known “overly-stiff” phenomenon existing in the standard FEM, and hence significantly improve the accuracy of the solution for the coupled systems. Intensive numerical studies have been conducted to verify the effectiveness of the coupled ES-/FS-FEM for structural-acoustic problems.