High-resolution compact numerical method for the system of 2D quasi-linear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications

In this paper, we present a novel approach to attain fourth-order approximate solution of 2D quasi-linear elliptic partial differential equation on an irrational domain. In this approach, we use nine grid points with dissimilar mesh in a single compact cell. We also discuss appropriate fourth-order numerical methods for the solution of the normal derivatives on a dissimilar mesh. The method has been protracted for solving system of quasi-linear elliptic equations. The convergence analysis is discussed to authenticate the proposed numerical approximation. On engineering applications, we solve various test problems, such as linear convection–diffusion equation, Burgers’equation, Poisson equation in singular form, NS equations, bi- and tri-harmonic equations and quasi-linear elliptic equations to show the efficiency and accuracy of the proposed methods. A comprehensive comparative computational experiment shows the accuracy, reliability and credibility of the proposed computational approach.

     

A Black-Box Iterative Solver Based on a Two-Level Schwarz Method

We propose a black-box parallel iterative method suitable for solving both elliptic and certain non-elliptic problems discretized on unstructured meshes. The method is analyzed in the case of the second order elliptic problems discretized on quasiuniform P1 and Q1 finite element meshes. The numerical experiments confirm the validity of the proved convegence estimate and show that the method can successfully be used for more difficult problems (e.g. plates, shells and Helmholtz equation in high-frequency domain.) Received: July 28, 1997; revised June 20, 1999     

Application of Post’s formula to optical pulse propagation in dispersive media

In this paper we consider the propagation of optical pulses in dielectric media with nontrivial dispersion relations. In particular, we implement Post’s Laplace transform formula to invert in time the Fourier–Laplace space coefficients which arise from the joint space solution of the optical dispersive wave equation. Due to the inefficiency of a direct application of this formula, we have considered and present here two more efficient implementations. In the first, the Gaver–Post method, we utilize the well known Gaver functionals but store intermediate calculations to improve efficiency. The second, the Bell–Post method, involves an analytic reformulation of Post’s formula such that knowledge of the dispersion relation and its derivatives are sufficient to invert the coefficients from Laplace space to time. Unlike other approaches to the dispersive wave equation which utilize a Debye–Lorentzian assumption for the dispersion relation, our algorithm is applicable to general Maxwell-Hopkinson dielectrics. Moreover, we formulate the approach in terms of the Fourier–Laplace coefficients which are characteristic of a dispersive medium but are independent of the initial pulse profile. They thus can be precomputed and utilized when necessary in a real-time system. Finally, we present an illustration of the method applied to optical pulse propagation in a free space and in two materials with Cole-type dispersion relations, room temperature ionic liquid (RTIL) hexafluorophosphate and brain white matter. From an analysis of these examples, we find that both methods perform better than a standard Post-formula implementation. The Bell–Post method is the more robust of the two, while the Gaver–Post is more efficient at high precision and Post formula approximation orders.     

Alternating Schwarz methods for partial differential equation-based mesh generation

To solve boundary value problems with moving fronts or sharp variations, moving mesh methods can be used to achieve reasonable solution resolution with a fixed, moderate number of mesh points. Such meshes are obtained by solving a nonlinear elliptic differential equation in the steady case, and a nonlinear parabolic equation in the time-dependent case. To reduce the potential overhead of adaptive partial differential equation-(PDE) based mesh generation, we consider solving the mesh PDE by various alternating Schwarz domain decomposition methods. Convergence results are established for alternating iterations with classical and optimal transmission conditions on an arbitrary number of subdomains. An analysis of a colouring algorithm is given which allows the subdomains to be grouped for parallel computation. A first result is provided for the generation of time-dependent meshes by an alternating Schwarz algorithm on an arbitrary number of subdomains. The paper concludes with numerical experiments illustrating the relative contraction rates of the iterations discussed.     

Finite element analysis of a compressible dynamic viscoelastic sphere using FFT

An extension to a compressible dynamic viscoelastic hollow sphere problem with both finite and infinite outer radius is performed. The governing viscoelastic equations of motion are transformed into the Laplace domain via the elastic–viscoelastic correspondence principle. Real and imaginary parts of the nodal displacements are obtained by solving a non-symmetric matrix equation in the complex Laplace domain. Inversion into the time domain is performed using the discrete inverse Fourier transform. Use is made of an infinite element in the infinite sphere problem. Numerical solutions are compared to both the exact Laplace and time domain solutions wherever possible.     

The finite superelement method and its application for solving science and technology problems

The algorithm for the construction and study of Fedorenko’s finite superelements method (FSEM) is described. Different variants of Fedorenko’s FSEM for simulation of media with small inclusions are presented. The algorithms are implemented and a software complex for the numerical solution of boundary value problems with singularities is developed. The theoretical substantiation of different variants of the method for elliptic equations on the example of the Laplace equation is implemented. The results of the numerical solution for several tasks are presented.     

Analysis of non-Fickian diffusion problems in a composite medium

The one-dimensional non-Fickian diffusion problems in a two-layered composite medium for finite and semi-infinite geometry are analyzed by using a hybrid application of the Laplace transform technique and control-volume method in conjunction with the hyperbolic shape functions, where the effect of the potential field is taken into account. The Laplace transform method used to remove the time-dependent terms in the governing differential equation and boundary conditions, and then the transformed equations are discretized by the control volume scheme. To evidence the accuracy of the present numerical method, a comparison between the present numerical results and analytical solution is made for the constant potential gradient. Results show that the present numerical results are accurate for various values of the potential gradient, relaxation time ratio, and diffusion coefficient ratio. It can be found that these values play an important role in the present problem. An interesting finding is that when the mass wave encounters an interface of the dissimilar materials, a portion of the wave is reflected and the rest is transmitted. The speed of propagation can change owing to the penetration of the mass wave into the region of the different material. The wave nature is significant only for short times and quickly dissipates with time.     

Spectral multidomain technique with Local Fourier Basis

A novel domain decomposition method for spectrally accurate solutions of PDEs is presented. A Local Fourier Basis technique is adapted for the construction of the elemental solutions in subdomains.C ¹ continuity is achieved on the interfaces by a matching procedure using the analytical homogeneous solutions of a one dimensional equation. The method can be applied to the solution of elliptic problems of the Poisson or Helmholtz type as well as to time discretized parabolic problems in one or more dimensions. The accuracy is tested for several stiff problems with steep solutions.The present domain decomposition approach is particularly suitable for parallel implementations, in particular, on MIMD type parallel machines.This research is supported partly by a grant from the French-Israeli Binational Foundation for 1991–1992.     

Finding Multiple Solutions to Elliptic PDE with Nonlinear Boundary Conditions

In this paper, in order to solve an elliptic partial differential equation with a nonlinear boundary condition for multiple solutions, the authors combine a minimax approach with a boundary integral-boundary element method, and identify a subspace and its special expression so that all numerical computation and analysis can be carried out more efficiently based on information of functions only on the boundary. Some mathematical justification of the new approach is established. An efficient and reliable local minimax-boundary element method is developed to numerically search for solutions. Details on implementation of the algorithm are also addressed. The existence and multiplicity of solutions to the problem are established under certain regular assumptions. Some conditions related to convergence of the algorithm and instability of solutions found by the algorithm are verified. To illustrated the method, numerical multiple solutions to some examples on domains with different geometry are displayed with their profile and contour plots.     

Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary

The aim of the paper is to design high-order artificial boundary conditions for the Schrödinger equation on unbounded domains in parallel with a treatment of the heat equation. We first introduce a circular artificial boundary to divide the unbounded definition domain into a bounded computational domain and an unbounded exterior domain. On the exterior domain, the Laplace transformation in time and Fourier series in space are applied to achieve the relation of special functions. Then the rational functions are used to approximate the relation of the special functions. Applying the inverse Laplace transformation to a series of simple rational function, we finally obtain the corresponding high-order artificial boundary conditions, where a sequence of auxiliary variables are utilized to avoid the high-order derivatives in respect to time and space. Furthermore, the finite difference method is formulated to discretize the reduced initial–boundary value problem with high-order artificial boundary conditions on a bounded computational domain. Numerical experiments are presented to illustrate the performance of our method.     

有限差分法的并行化计算实现

有限差分法是求解偏微分方程近似解的一种重要的数值方法。并行化计算可提高复杂计算问题的效率,二维场中拉普拉斯方程的差分格式非常适合并行化方法的计算。如何将串行部分并行化以提高大规模计算的效率,MPI（消息传递接口）是实现并行程序设计的标准之一。虚拟进程（MPI_PROC_NULL）是MPI中的假想进程,它的引用可简化MPI编程中的通信部分,引入虚拟进程编写代码,可实现有限差分方法的并行化计算。     

Dempster–Shafer evidential theory for the automated selection of parameters for Talbot’s method contours and application to matrix exponentiation

In this paper, the Dempster–Shafer theory of evidential reasoning is applied to the problem of optimal contour parameters selection in Talbot’s method for the numerical inversion of the Laplace transform. The fundamental concept is the discrimination between rules for the parameters that define the shape of the contour based on the features of the function to invert. To demonstrate the approach, it is applied to the computation of the matrix exponential via numerical inversion of the corresponding resolvent matrix. Training for the Dempster–Shafer approach is performed on random matrices. The algorithms presented have been implemented in MATLAB. The approximated exponentials from the algorithm are compared with those from the rational approximation for the matrix exponential returned by the MATLAB expm function.     

Dimension splitting algorithm for a three-dimensional elliptic equation

This paper presents a finite-element dimension splitting algorithm (DSA) for a three-dimensional (3D) elliptic equation in a cubic domain. The main idea of DSA is that a 3D elliptic equation can be transformed into a series of two-dimensional (2D) elliptic equations in the X–Y plane along the Z-direction. The convergence speed of the DSA for a 3D elliptic equation depends mainly on the mesh scale of the Z-direction. P ₂ finite-element discretization in the Z-direction for DSA is adopted to accelerate the convergence speed of DSA. The error estimates are given for DSA applying P ₁ or P ₂ finite-element discretization in the Z-direction. Finally, some numerical examples are presented. We apply the parallel solving technology to our numerical examples and obtain good parallel efficiency. These numerical experiments test and verify theoretical results.     

A Closed-Form Formula for the RBF-Based Approximation of the Laplace–Beltrami Operator

In this paper we present a method that uses radial basis functions to approximate the Laplace–Beltrami operator that allows to solve numerically diffusion (and reaction–diffusion) equations on smooth, closed surfaces embedded in \(\mathbb {R}^3\). The novelty of the method is in a closed-form formula for the Laplace–Beltrami operator derived in the paper, which involve the normal vector and the curvature at a set of points on the surface of interest. An advantage of the proposed method is that it does not rely on the explicit knowledge of the surface, which can be simply defined by a set of scattered nodes. In that case, the surface is represented by a level set function from which we can compute the needed normal vectors and the curvature. The formula for the Laplace–Beltrami operator is exact for radial basis functions and it also depends on the first and second derivatives of these functions at the scattered nodes that define the surface. We analyze the converge of the method and we present numerical simulations that show its performance. We include an application that arises in cardiology.     

Coupling of two-dimensional hyperbolic and elliptic equations

We deal with the problem of interfacing an elliptic equation with a hyperbolic one in adjoining two-dimensional domains. We provide proper interface conditions and propose an iterative algorithm that alternates the solution of the elliptic equation and of the hyperbolic one within the respective subdomains. The spectral Chebyshev collocation method is used to discretize the result subproblems. Several numerical tests show that the iterative algorithm is very effective and that the computed solutions are quite accurate. This approach can be successfully applied to computational fluid dynamics problems, such as the simulation of flow around a body.     

Higher-order schemes for the Laplace transformation method for parabolic problems

In this paper we solve linear parabolic problems using the three stage noble algorithms. First, the time discretization is approximated using the Laplace transformation method, which is both parallel in time (and can be in space, too) and extremely high order convergent. Second, higher-order compact schemes of order four and six are used for the the spatial discretization. Finally, the discretized linear algebraic systems are solved using multigrid to show the actual convergence rate for numerical examples, which are compared to other numerical solution methods.     

Highly Scalable Two- and Three-Dimensional Navier-Stokes Parallel Solvers on MIMD Multiprocessors

In this paper we present a new parallel algorithm for the solution of the incompressible two- and three-dimensional Navier-Stokes equations. The parallelization is achieved via domain decomposition. The computational region is considered in the form of a 2-D or 3-D periodic box decomposed into parallel strips (slabs). For time discretization we use a third order multistep method of [11]. The time discretization procedure results in solving global elliptic problems of (monotonic) Helmholtz and Poisson types in each time step. For the space discretization we employ the multidomain local Fourier (MDLF) method that was developed in [9, 10, 13]. The discretization in the periodic directions is performed by the standard Fourier method. In the direction across the strips we use the Local Fourier Basis technique which involves the overlapping of the neighboring subdomains and smoothing of local functions across the interior boundaries (interfaces). The matching of the local solutions is performed by adding properly weighted interface Green's functions. Their amplitudes are found in terms of the jumps of the solution and its first derivatives at the interfaces. The present paper extends the results of our previous works [1, 9, 10, 13] on parallel use of the MDLF method in three-fold aspects: 1. In [1] a model Navier-Stokes type system was considered which does not include the pressure term. Correspondingly, in each time step only the Helmholtx type equations were solved. It was shown that the parallel solution of this equation can be accomplished using only local (neighbor-to-neighbor) communication due to localization properties of the Helmholtz operator. We consider the complete Navier-Stokes system including the pressure term. The solution of the Poisson equation for pressure has the potential to degrade the performance and the achieved speedup of a parallel algorithm due to the global nature of this equation that necessitates global communication among the processors. However, we show that only a few lowest harmonics require for the global data transfer whereas the rest of harmonics can be treated locally. Therefore, most of the communication that is required for parallelization of the Navier-Stokes solver using the MDLF method is mainly local between adjacent subdomains (processors). Moreover, the percentage of the time spent in global communication reduces as the size of the problem increases. Thus, the present parallel algorithm is highly scalable. 2. In [l] we considered only 2-D equations. In this paper we extend the previous technique to 3-D problems. 3. Previously, the MDLF solver was implemented only on the MEIKO parallel machine. In this paper the 2-D and 3-D Navier-Stokes solvers are implemented on three MIMD message-passing multiprocessors (a 60-processors IBM SP2, a 20-processors MOSIX [3], and a network of 10 Alpha workstations) and achieve an efficiency of more than 70% to 95%. The same code written with the PVM (parallel virtual machine [7]) software package was executed on all the above distinct computational platforms. Detailed performance results, which include scalability analysis, are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Solution of Dirichlet problem for a rectangular region in terms of elliptic functions

In this study, the Green function of the (interior) Dirichlet problem for the Laplace (also Poisson) differential equation in a rectangular domain is expressed in terms of elliptic functions and the solution of the problem is based on the Green function and therefore on the elliptic functions. The method of solution for the Dirichlet problem by the Green function is presented; the Green function and transformation required for the solution of the Dirichlet problem in the rectangular region is found and the problem is solved in the rectangular region. An example for the problem in the rectangular region is given in order to present an application of the solution of Dirichlet problem. The equation is solved first by the known method of separation of variables and then in terms of elliptic functions; the results of both methods are compared. The results are found to be consistent but the advantage of this method is that the solution is obtained in terms of elementary functions.     

核电站中椭圆柱形屏蔽体辐射衰减率的仿真研究

为了预测核电站中椭圆柱形屏蔽体的辐射衰减率，该文用数值解析法对放射线源到评价点的放射线所穿透椭圆柱形屏蔽物体的交点进行了研究。首先用空间解析法建立了椭圆柱形物体的数学模型。然后解曲面方程求出交点。在判断放射线是否穿透此椭圆柱体时，该文又提出了椭圆柱形屏蔽体的假想外接长方体的建模方法，将曲面方程联立解的复杂计算简化为一次代数方程的简单计算，大幅度地缩短了计算时间。