Kinetic flux vector splitting for radiation hydrodynamical equations

This paper is interested in the kinetic flux vector splitting (KFVS) for the multidimensional radiation hydrodynamical equations (RHEs) in zero diffusion limit. First, a generalized Maxwell–Boltzmann distribution function with two new parameters of temperature approximation is introduced to recover the macroscopic equations. These parameters are uniquely determined by macroscopic variables. Then, a high resolution KFVS method is proposed for the solution of the multidimensional RHEs. It does not require any Riemann solvers. Finally, several numerical examples are given to show the performance of our scheme.     

一致高精度KFVS方法用于多分量流的计算

５１．引言 很多传统的守恒型差分格式用于多组分流体的数值计算时,如果比热比１在不同流体间的界面附近不为常数,则数值解容易产生数值误差,并可能导致非物理解．文［１,４,７］就一些具体的格式提出了相应的减少物质界面附近数值误差的处理方法．它们的主要思想是对原来的算法作相应的非守恒校正, Ｋａｒｎｉ在文［５］中使用了原始变量算法求解多组分流,在文［６１又进一步研究了原始变量方法和 Ｌｅｖｅｌ Ｓｅｔ（位标）方法混合的算法．董素琴等［’］研究了多组分流体的二维非守恒型差分格式,结果表明,计算解在界面附近的误…     

Numerical Modeling of Degenerate Equations in Porous Media Flow

In this paper is introduced a new numerical formulation for solving degenerate nonlinear coupled convection dominated parabolic systems in problems of flow and transport in porous media by means of a mixed finite element and an operator splitting technique, which, in turn, is capable of simulating the flow of a distinct number of fluid phases in different porous media regions. This situation naturally occurs in practical applications, such as those in petroleum reservoir engineering and groundwater transport. To illustrate the modelling problem at hand, we consider a nonlinear three-phase porous media flow model in one- and two-space dimensions, which may lead to the existence of a simultaneous one-, two- and three-phase flow regions and therefore to a degenerate convection dominated parabolic system. Our numerical formulation can also be extended for the case of three space dimensions. As a consequence of the standard mixed finite element approach for this flow problem the resulting linear algebraic system is singular. By using an operator splitting combined with mixed finite element, and a decomposition of the domain into different flow regions, compatibility conditions are obtained to bypass the degeneracy in order to the degenerate convection dominated parabolic system of equations be numerically tractable without any mathematical trick to remove the singularity, i.e., no use of a parabolic regularization. Thus, by using this procedure, we were able to write the full nonlinear system in an appropriate way in order to obtain a nonsingular system for its numerical solution. The robustness of the proposed method is verified through a large set of high-resolution numerical experiments of nonlinear transport flow problems with degenerating diffusion conditions and by means of a numerical convergence study.     

A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation

An efficient numerical technique is proposed to solve one- and two-dimensional space fractional tempered fractional diffusion-wave equations. The space fractional is based on the Riemann–Liouville fractional derivative. At first, the temporal direction is discretized using a second-order accurate difference scheme. Then a classic Galerkin finite element is employed to obtain a full-discrete scheme. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique.     

Meshless kinetic upwind method for compressible, viscous rotating flows

This paper presents the split-stencil least square kinetic upwind method for Navier–Stokes (SLKNS) solver using kinetic flux vector splitting (KFVS) scheme with Chapman-Enskog distribution. SLKNS solver operates on an arbitrary distribution of points and uses a novel least squares method which differs from the normal equations approach as it generates the non-symmetric cross-product matrix by selective splitting of the set of neighbours to avoid ill-conditioning. SLKNS also uses the axi-symmetric formulation of the Boltzmann equation and kinetic slip boundary condition. SLKNS is capable of capturing weak secondary flows as well as features of strong rotation characterized by steep density gradient and thin boundary layers towards the peripheral region with a rarefied central core.     

Modified kinetic flux vector splitting (m-KFVS) method for compressible flows

To resolve many flow features accurately, like accurate capture of suction peak in subsonic flows and crisp shocks in flows with discontinuities, to minimise the loss in stagnation pressure in isentropic flows or even flow separation in viscous flows require an accurate and low dissipative numerical scheme. The first order kinetic flux vector splitting (KFVS) method has been found to be very robust but suffers from the problem of having much more numerical diffusion than required, resulting in inaccurate computation of the above flow features. However, numerical dissipation can be reduced by refining the grid or by using higher order kinetic schemes. In flows with strong shock waves, the higher order schemes require limiters, which reduce the local order of accuracy to first order, resulting in degradation of flow features in many cases. Further, these schemes require more points in the stencil and hence consume more computational time and memory. In this paper, we present a low dissipative modified KFVS (m-KFVS) method which leads to improved splitting of inviscid fluxes. The m-KFVS method captures the above flow features more accurately compared to first order KFVS and the results are comparable to second order accurate KFVS method, by still using the first order stencil.     

Lagrange–Galerkin method for unsteady free surface water waves

We present a new numerical technique to approximate solutions to unsteady free surface flows modelled by the two-dimensional shallow water equations. The method we propose in this paper consists of an Eulerian–Lagrangian splitting of the equations along the characteristic curves. The Lagrangian stage of the splitting is treated by a non-oscillatory modified method of characteristics, while the Eulerian stage is approximated by an implicit time integration scheme using finite element method for spatial discretization. The combined two stages lead to a Lagrange–Galerkin method which is robust, second order accurate, and simple to implement for problems on complex geometry. Numerical results are shown for several test problems with different ranges of difficulty.     

Unconditionally positivity and boundedness preserving schemes for a FitzHugh–Nagumo equation

In this paper, a semi-explicit scheme is constructed for the space-independent FitzHugh–Nagumo equation. Qualitative stability analysis shows that the semi-explicit scheme is dynamically consistent with the space independent equation. Then, the semi-explicit scheme is extended to construct a new finite difference scheme for the full FitzHugh–Nagumo equation in one- and two-space dimensions, respectively. According to the theory of M-matrices, it is proved that these new schemes are able to preserve the positivity and boundedness of solutions of the corresponding equations for arbitrary step sizes. The consistency and numerical stability of these schemes is also analysed. Combined with the property of the strictly diagonally dominant matrix, the convergence of these schemes is established. Numerical experiments illustrate our results and display the advantages of our schemes in comparison to some other schemes.     

一类二维粘性波动方程的交替方向有限体积元方法

针对二维粘性波动方程模型问题,提出了一类基于双线性插值的交替方向有限体积元方法,并给出了两种具体计算格式,一是基于有限差分方法中Douglas思想的格式,二是一类推广型的局部一维格式.分析证明了该方法按照L~2范数在时间和空间方向均有二阶收敛精度.最后,数值算例验证了算法的有效性和精确性.     

Nonlinear transient phenomena in saturated porous media

In this report the transient ‘static’ and ‘dynamic’ response of saturated porous medium is analyzed. The saturated porous medium viewed as a two-phase system whose state is described by the stresses, displacements, velocities and accelerations within each phase. The coupled field and constitutive equations based on mixture's theories are presented, and solved numerically by the use of the finite element method. Time integration is achieved by using an implicit/explicit predictor/multicorrector scheme developed by Hughes and co-workers. The procedure allows for a convenient selection of implicit and explicit elements, and for an implicit/explicit split of the various operators appearing in the differential equations. Accuracy and versatility of the proposed procedure are demonstrated by applying it to obtain solution to various dynamic and pseudo-static, one- and two-dimensional initial value problems.     

Numerical Simulation of Microflows Using Moment Methods with Linearized Collision Operator

Hyperbolic moment equations based on Burnett’s expansion of the distribution function are derived for the Boltzmann equation with linearized collision operator. Boundary conditions are equipped for these models, and it is proven that the number of boundary conditions is correct for a large class of moment models. A new second-order numerical scheme is proposed for solving these moment equations, and the new method is suitable for both ordered- and full-moment theories. Numerical experiments are carried out for both one- and two-dimensional problems to show the performance of the moment methods.     

A finite difference real ghost fluid method on moving meshes with corner-transport upwind interpolation

The real ghost fluid method (RGFM) [Wang CW, Liu TG, Khoo BC. A real-ghost fluid method for the simulation of multi-medium compressible flow. SIAM J Sci Comput 2006;28:278–302] has been shown to be more robust than previous versions of GFM for simulating multi-medium flow problems with large density and pressure jumps. In this paper, a finite difference RGFM is combined with adaptive moving meshes for one- and two-dimensional problems. A high resolution corner-transport upwind (CTU) method is used to interpolate approximate solutions from old quadrilateral meshes to new ones. Unlike the dimensional splitting interpolation, the CTU method takes into account the transport across corner points, which is physically more sensible. Several one- and two-dimensional examples with large density and pressure jumps are computed. The results show the present moving mesh method can effectively reduce the conservative errors produced by GFM and can increase the computational efficiency.     

Fractional diffusion equations by the Kansa method

This study makes the first attempt to apply the Kansa method in the solution of the time fractional diffusion equations, in which the MultiQuadrics and thin plate spline serve as the radial basis function. In the discretization formulation, the finite difference scheme and the Kansa method are respectively used to discretize time fractional derivative and spatial derivative terms. The numerical solutions of one- and two-dimensional cases are presented and discussed, which agree well with the corresponding analytical solution.     

H2 regularity properties of singular parameterizations in isogeometric analysis

Isogeometric analysis (IGA) is a numerical simulation method which is directly based on the NURBS-based representation of CAD models. It exploits the tensor-product structure of 2- or 3-dimensional NURBS objects to parameterize the physical domain. Hence the physical domain is parameterized with respect to a rectangle or to a cube. Consequently, singularly parameterized NURBS surfaces and NURBS volumes are needed in order to represent non-quadrangular or non-hexahedral domains without splitting, thereby producing a very compact and convenient representation.The Galerkin projection introduces finite-dimensional spaces of test functions in the weak formulation of partial differential equations. In particular, the test functions used in isogeometric analysis are obtained by composing the inverse of the domain parameterization with the NURBS basis functions. In the case of singular parameterizations, however, some of the resulting test functions do not necessarily fulfill the required regularity properties. Consequently, numerical methods for the solution of partial differential equations cannot be applied properly.We discuss the regularity properties of the test functions. For one- and two-dimensional domains we consider several important classes of singularities of NURBS parameterizations. For specific cases we derive additional conditions which guarantee the regularity of the test functions. In addition we present a modification scheme for the discretized function space in case of insufficient regularity. It is also shown how these results can be applied for computational domains in higher dimensions that can be parameterized via sweeping.     

Generalization of the CABARET scheme to two-dimensional orthogonal computational grids

It is proposed to generalize the CABARET method to a two-dimensional system of Euler equations. The transition from one-dimensional problems of gas dynamics to multidimensional ones for the CABARET scheme involves a number of innovative aspects. The first is a procedure for spatial splitting of an algorithm in order to calculate new flow variables. The second one is the specific application of the maximum principle aimed at regularizing the solutions for inhomogeneous equations of transfer of local invariants in different directions. Examples are given of test and model calculations.     

Adaptive dimension splitting algorithm for three-dimensional elliptic equations

An adaptive dimension splitting algorithm for three-dimensional (3D) elliptic equations is presented in this paper. We propose residual and recovery-based error estimators with respect to X?Y plane direction and Z direction, respectively, and construct the corresponding adaptive algorithm. Two-sided bounds of the estimators guarantee the efficiency and reliability of such error estimators. Numerical examples verify their efficiency both in estimating the error and in refining the mesh adaptively. This algorithm can be compared with or even better than the 3D adaptive finite element method with tetrahedral elements in some cases. What is more, our new algorithm involves only two-dimensional mesh and one-dimensional mesh in the process of refining mesh adaptively, and it can be implemented in parallel.     

A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients

In this paper, a combined compact finite difference method (CCD) together with alternating direction implicit (ADI) scheme is developed for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients. The proposed CCD-ADI method is second-order accurate in time variable and sixth-order accurate in space variable. For the linear hyperbolic equation, the CCD-ADI method is shown to be unconditionally stable by using the Von Neumann stability analysis. Numerical results for both linear and nonlinear hyperbolic equations are presented to illustrate the high accuracy of the proposed method.     

A sixth-order finite difference WENO scheme for Hamilton–Jacobi equations

In this paper, a sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed to approximate the viscosity solution of the Hamilton–Jacobi equations. This new WENO scheme has the same spatial nodes as the classical fifth-order WENO scheme proposed by Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM. J. Sci. Comput. 21 (2000), pp. 2126–2143] but can be as high as sixth-order accurate in smooth region while keeping sharp discontinuous transitions with no spurious oscillations near discontinuities. Extensive numerical experiments in one- and two-dimensional cases are carried out to illustrate the capability of the proposed scheme.     

Primal Discontinuous Galerkin Methods for Time-Dependent Coupled Surface and Subsurface Flow

This paper introduces and analyzes a numerical method based on discontinuous finite element methods for solving the two-dimensional coupled problem of time-dependent incompressible Navier-Stokes equations with the Darcy equations through Beaver-Joseph-Saffman’s condition on the interface. The proposed method employs Crank-Nicolson discretization in time (which requires one step of a first order scheme namely backward Euler) and primal DG method in space. With the correct assumption on the first time step optimal error estimates are obtained that are high order in space and second order in time.     

A new ADI technique for two-dimensional parabolic equation with an integral condition

Nonclassical parabolic initial-boundary value problems arise in the study of several important physical phenomena. This paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions. A new fourth-order finite difference technique, based upon the Noye and Hayman (N-H) alternating direction implicit (ADI) scheme, is used as the basis to solve the two-dimensional time dependent diffusion equation with an integral condition replacing one boundary condition. This scheme uses less central processor time (CPU) than a second-order fully implicit scheme based on the classical backward time centered space (BTCS) method for two-dimensional diffusion. It also has a larger range of stability than a second-order fully explicit scheme based on the classical forward time centered space (FTCS) method. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments for the new method are presented. The central processor times needed are also reported. Error estimates derived in the maximum norm are tabulated.