The Immersed Interface Method for Simulating Two-Fluid Flows

We develop the immersed interface method (IIM) to simulate a two-fluid flow of two immiscible fluids with different density and viscosity. Due to the surface tension and the discontinuous fluid properties, the two-fluid flow has nonsmooth velocity and discontinuous pressure across the moving sharp interface separating the two fluids. The IIM computes the flow on a fixed Cartesian grid by incorporating into numerical schemes the necessary jump conditions induced by the interface. We present how to compute these necessary jump conditions from the analytical principal jump conditions derived in [Xu, DCDS, Supplement 2009, pp. 838-845]. We test our method on some canonical two-fluid flows. The results demonstrate that the method can handle large density and viscosity ratios, is second-order accurate in the infinity norm, and conserves mass inside a closed interface.     

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

Based on Li’s immersed interface method (IIM), an ADI-type finite difference scheme is proposed for solving two-dimensional nonlinear convection–diffusion interface problems on a fixed cartesian grid, which is unconditionally stable and converges with two-order accuracy in both time and space in maximum norm. Correction terms are added to the right-hand side of standard ADI scheme at irregular points. The nonlinear convection terms are treated by Adams–Bashforth method, without affecting the stability of difference schemes. A new method for computing the correction terms is developed, in which the Adams–Bashforth method is employed. Thus we can get an explicit approximation for the computation of corrections, when the jump condition is solution-dependent. Three numerical experiments are displayed and analyzed. The numerical results show good agreement with the exact solutions and confirm the convergence order.     

Opposing flows in a one dimensional convection-diffusion problem

In this paper, we examine a particular class of singularly perturbed convection-diffusion problems with a discontinuous coefficient of the convective term. The presence of a discontinuous convective coefficient generates a solution which mimics flow moving in opposing directions either side of some flow source. A particular transmission condition is imposed to ensure that the differential operator is stable. A piecewise-uniform Shishkin mesh is combined with a monotone finite difference operator to construct a parameter-uniform numerical method for this class of singularly perturbed problems.     

Boundary‐conforming coordinates with grid line control for acoustic scattering from complexly shaped obstacles

The current work sets forth a practical approach to numerically solve two‐dimensional direct acoustic scattering problems from complexly shaped scatterers with severe singularities, such as corners and cusps. First, boundary conforming coordinates are generated. This generation is performed through an elliptic grid generator algorithm, including control of the coordinate lines. The grid line control solely depends on the initial distribution of grid points. Following the grid generation process, the initial boundary value problem, modelling the scattering phenomenon, is formulated in terms of the new curvilinear coordinates, and a finite‐difference time domain method is implemented. The presence of the boundary singularities causes instability of the numerical method. However, by appropriately controlling the distance between grid lines in the vicinity of these singularities, stability and convergence are achieved. A semianalytical formula for the differential scattering cross‐section is obtained from the discrete Fourier transform of the computed scattered pressure field. The method is successfully applied to several interesting scatterers of various shapes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

High-Order Finite-Differences Schemes to Solve Poisson's Equation in Polar Coordinates

In the present work finite difference schemes of second andfourth order are derived for the solution of Poisson's equationin polar coordinates. To solve the resulting system of linearequations, a direct method similar to Hockney's method is developed.The schemes are tested on six test problems whose exact solutionsare known. The numerical results obtained by these finite-differenceschemes show that they produce very accurate results.     

基于最小二乘法的球形传递装置扩散优化控制问题求解

针对物流工程领域学科所涉及的扩散方程中的扩散系数求解问题,建立了球形传递装置扩散优化控制模型.首先,利用球形坐标系变换公式,得到极坐标下的扩散优化控制模型.然后,采用迭代的方法,通过最小二乘法估计该模型的扩散系数.最后,通过数值实例,验证了扩散优化控制模型及算法的有效性和收敛性.     

Two-Level preconditioned Krylov subspace methods for the solution of three-dimensional heterogeneous Helmholtz problems in seismics

In this paper we address the solution of three-dimensional heterogeneous Helmholtz problems discretized with compact fourth-order finite difference methods with application to acoustic waveform inversion in geophysics. In this setting, the numerical simulation of wave propagation phenomena requires the approximate solution of possibly very large linear systems of equations. We propose an iterative two-grid method where the coarse grid problem is solved inexactly. A single cycle of this method is used as a variable preconditioner for a flexible Krylov subspace method. Numerical results demonstrate the usefulness of the algorithm on a realistic three-dimensional application. The proposed numerical method allows us to solve wave propagation problems with single or multiple sources even at high frequencies on a reasonable number of cores of a distributed memory cluster.     

Single cell finite difference approximations of O(kh2 + h4) for ∂u/∂x for one space dimensional nonlinear parabolic equation

We report a new two‐level explicit finite difference method of O(kh² + h⁴) using three spatial grid points for the numerical solution of for the solution of one‐space dimensional nonlinear parabolic partial differential equation subject to appropriate initial and Dirichlet boundary conditions. The method is shown to be unconditionally stable when applied to a linear equation. The proposed method is applicable to the problems both in cartesian and polar coordinates. Numerical examples are provided to demonstrate the efficiency and accuracy of the method discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 408–415, 2000     

An Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems

A new fast algorithm based on the augmented immersed interface method and a fast Poisson solver is proposed to solve three dimensional elliptic interface problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable which should be chosen such that the original flux jump condition is satisfied. The discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing PDEs in a neighborhood of control points on the interface. The interpolation scheme is the key to the success of the augmented IIM particularly. In this paper, the key new idea is to select interpolation points along the normal direction in line with the flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The number of the GMRES iterations is independent of the mesh size.     

Finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains

A finite difference time-dependent numerical method for the wave equation, supported by recently derived novel elliptic grids, is analyzed. The method is successfully applied to single and multiple two-dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries. The new grids have nearly uniform cell area (J-grids) and nearly uniform grid line spacing (αγ-grids). Numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state. The restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing. As a consequence, moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost. Also, numerical solutions for wave problems inside annular regions of complex shapes are obtained. The use of the new grids results in late time stability in contrast with other classical finite difference time-dependent methods.     

An interpolation matched interface and boundary method for elliptic interface problems

An interpolation matched interface and boundary (IMIB) method with second-order accuracy is developed for elliptic interface problems on Cartesian grids, based on original MIB method proposed by Zhou et al. [Y. Zhou, G. Wei, On the fictious-domain and interpolation formulations of the matched interface and boundary method, J. Comput. Phys. 219 (2006) 228-246]. Explicit and symmetric finite difference formulas at irregular grid points are derived by virtue of the level set function. The difference scheme using IMIB method is shown to satisfy the discrete maximum principle for a certain class of problems. Rigorous error analyses are given for the IMIB method applied to one-dimensional (1D) problems with piecewise constant coefficients and two-dimensional (2D) problems with singular sources. Comparison functions are constructed to obtain a sharp error bound for 1D approximate solutions. Furthermore, we compare the ghost fluid method (GFM), immersed interface method (IIM), MIB and IMIB methods for 1D problems. Finally, numerical examples are provided to show the efficiency and robustness of the proposed method.     

A difference scheme for a conjugate-operator model of a heat conduction problem in polar coordinates

In polar coordinates, a discrete analog of the conjugate-operator model of a heat conduction problem is formulated to hold the structure of the original model. The difference scheme converges with second-order accuracy in the case of discontinuous parameters of the medium in the Fourier law and irregular grids. An efficient algorithm for solving the discrete conjugate-operator model when heat conduction tensor is a unit operator is proposed.     

Application of the multigrid approach for solving 3D Navier-Stokes equations on hexahedral grids using the discontinuous Galerkin method

The Galerkin method with discontinuous basis functions is adapted for solving the Euler and Navier-Stokes equations on unstructured hexahedral grids. A hybrid multigrid algorithm involving the finite element and grid stages is used as an iterative solution method. Numerical results of calculating the sphere inviscid flow, viscous flow in a bent pipe, and turbulent flow past a wing are presented. The numerical results and the computational cost are compared with those obtained using the finite volume method.     

Numerical modelling of free-surface shallow flows over irregular topography with complex geometry

A well-balanced Godunov-type finite volume algorithm is developed for modelling free-surface shallow flows over irregular topography with complex geometry. The algorithm is based on a new formulation of the classical shallow water equations in hyperbolic conservation form. Unstructured triangular grids are used to achieve the adaptability of the grid to the geometry of the problem and to facilitate localised refinement. The numerical fluxes are calculated using HLLC approximate Riemann solver, and the MUSCL-Hancock predictor–corrector scheme is adopted to achieve the second-order accuracy both in space and in time where the solutions are continuous, and to achieve high-resolution results where the solutions are discontinuous. The novelties of the algorithm include preserving well-balanced property without any additional correction terms and the wet/dry front treatments. The good performance of the algorithm is demonstrated by comparing numerical and theoretical results of several benchmark problems, including the preservation of still water over a two-dimensional hump, the idealised dam-break flow over a frictionless flat rectangular channel, the circular dam-break, and the shock wave from oblique wall. Besides, two laboratory dam-break cases are used for model validation. Furthermore, a practical application related to dam-break flood wave propagation over highly irregular topography with complex geometry is presented. The results show that the algorithm can correctly account for free-surface shallow flows with respect to its effectiveness and robustness thus has bright application prospects.     

Fast direct solver for Poisson equation in a 2D elliptical domain

In this article, we extend our previous work 3 for developing some fast Poisson solvers on 2D polar and spherical geometries to an elliptical domain. Instead of solving the equation in an irregular Cartesian geometry, we formulate the equation in elliptical coordinates. The solver relies on representing the solution as a truncated Fourier series, then solving the differential equations of Fourier coefficients by finite difference discretizations. Using a grid by shifting half mesh away from the pole and incorporating the derived numerical boundary value, the difficulty of coordinate singularity can be elevated easily. Unlike the case of 2D disk domain, the present difference equation for each Fourier mode is coupled with its conjugate mode through the numerical boundary value near the pole; thus, those two modes are solved simultaneously. Both second‐ and fourth‐order accurate schemes for Dirichlet and Neumann problems are presented. In particular, the fourth‐order accuracy can be achieved by a three‐point compact stencil which is in contrast to a five‐point long stencil for the disk case. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 72–81, 2004     

Solving the problem of non-stationary filtration of substance by the discontinuous Galerkin method on unstructured grids

A numerical algorithm is proposed for solving the problem of non-stationary filtration of substance in anisotropic media by the Galerkin method with discontinuous basis functions on unstructured triangular grids. A characteristic feature of this method is that the flux variables are considered on the dual grid. The dual grid comprises median control volumes around the nodes of the original triangular grid. The flux values of the quantities on the boundary of an element are calculated with the help of stabilizing additions. For averaging the permeability tensor over the cells of the dual grid, the method of support operators is applied. The method is studied on the example of a two-dimensional boundary value problem. The convergence and approximation of the numerical method are analyzed, and results of mathematical modeling are presented. The numerical results demonstrate the applicability of this approach for solving problems of non-stationary filtration of substance in anisotropic media by the discontinuous Galerkin method on unstructured triangular grids.     

A New Exponential Compact Scheme for the Two-Dimensional Unsteady Nonlinear Burgers' and Navier-Stokes Equations in Polar Cylindrical Coordinates

In this article, a new compact difference scheme is proposed in exponential form to solve two-dimensional unsteady nonlinear Burgers' and Navier-Stokes equations of motion in polar cylindrical coordinates by using half-step discretization. At each time level by using only nine grid points in space, the proposed scheme gives accuracy of order four in space and two in time. The method is directly applicable to the equations having singularities at boundary points. Stability analysis is explained in detail and many benchmark problems like Burgers', Navier-Stokes and Taylor-vortex problems in polar cylindrical coordinates are solved to verify the accuracy and efficiency of the scheme.     

Node-pair finite volume/finite element schemes for the Euler equation in cylindrical and spherical coordinates

A numerical scheme is presented for the solution of the compressible Euler equations in both cylindrical and spherical coordinates. The unstructured grid solver is based on a mixed finite volume/finite element approach. Equivalence conditions linking the node-centered finite volume and the linear Lagrangian finite element scheme over unstructured grids are reported and used to devise a common framework for solving the discrete Euler equations in both the cylindrical and the spherical reference systems. Numerical simulations are presented for the explosion and implosion problems with spherical symmetry, which are solved in both the axial–radial cylindrical coordinates and the radial–azimuthal spherical coordinates. Numerical results are found to be in good agreement with one-dimensional simulations over a fine mesh.     

<Emphasis Type="Italic">ε</Emphasis>-Uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with nterior layers

This article is devoted to the study of a hybrid numerical scheme for a class of singularly perturbed parabolic convection-diffusion problems with discontinuous convection coefficients. In general, the solutions of this class of problems possess strong interior layers. To solve these problems, we discretize the time derivative by the backward-Euler method and the spatial derivatives by a hybrid finite difference scheme (a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions) on a layer resolving piecewise-uniform Shishkin mesh. It is proved that the method converges uniformly in the discrete supremum norm with almost second-order spatial accuracy. Moreover, an optimal order of convergence (up to a logarithmic factor) is obtained inside the layer regions. Extensive numerical experiments are conducted to support the theoretical results and also, to demonstrate the accuracy of this method.     

Simplified immersed interface methods for elliptic interface problems with straight interfaces

In this article, we propose simplified immersed interface methods for elliptic partial/ordinary differential equations with discontinuous coefficients across interfaces that are few isolated points in 1D, and straight lines in 2D. For one‐dimensional problems or two‐dimensional problems with circular interfaces, we propose a conservative second‐order finite difference scheme whose coefficient matrix is symmetric and definite. For two‐dimensional problems with straight interfaces, we first propose a conservative first‐order finite difference scheme, then use the Richardson extrapolation technique to get a second‐order method. In both cases, the finite difference coefficients are almost the same as those for regular problems. Error analysis is given along with numerical example. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 188–203, 2012