An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a \(C^0\) finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.     

Dynamic analysis of thick laminated shell panel with piezoelectric layer based on three dimensional elasticity solution

Elasticity solution is presented for infinitely long, simply-supported, orthotropic, piezoelectric shell panel under dynamic pressure excitation. The direct and inverse piezoelectric effects are considered. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in circumferential direction. The resulting ordinary differential equations are solved by the finite element method. Numerical examples are presented for [0/90/P] lamination, where P indicates the piezoelectric layer. Finally the results are compared with the published results.     

One-step implicit methods or solving delay differential equations

Two one-step implicit methods—the second order Trapezium method and the fourth order implicit Runge-Kutta method for solving the delay differential equations (DDE) are developed. The significance of implicit methods lie in their 4-stability for ordinary differential equations. Different techniques are used to approximate the delay term. We also discuss the local truncation error estimate. Numerical examples are solved to show the effectiveness of the methods so developed.     

Two-grid quasilinearization approach to ODEs with applications to model problems in physics and mechanics

In this paper we propose a two-grid quasilinearization method for solving high order nonlinear differential equations. In the first step, the nonlinear boundary value problem is discretized on a coarse grid of size H. In the second step, the nonlinear problem is linearized around an interpolant of the computed solution (which serves as an initial guess of the quasilinearization process) at the first step. Thus, the linear problem is solved on a fine mesh of size h, h?H. On this base we develop two-grid iteration algorithms, that achieve optimal accuracy as long as the mesh size satisfies h=O(Hr²), r=1,2,… , where r is the rth Newton's iteration for the linearized differential problem. Numerical experiments show that a large class of NODEs, including the Fisher-Kolmogorov, Blasius and Emden-Fowler equations solving with two-grid algorithm will not be much more difficult than solving the corresponding linearized equations and at the same time with significant economy of the computations.     

ATOMFT: solving ODEs and DAEs using Taylor series

Taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long Taylor series. The series terms are generated recursively using the techniques of automatic differentiation. The ATOMFT system includes a translator to transform statements of the system of ODEs into a FORTRAN 77 object program that is compiled, linked with the ATOMFT runtime library, and run to solve the problem. We review the use of the ATOMFT system for nonstiff and stiff ODEs, the propagation of global errors, and applications to differential algebraic equations arising from certain control problems, to boundary value problems, to numerical quadrature, and to delay problems.     

Application of a posteriori error estimation to finite element simulation of incompressible Navier-Stokes flow

The main goal of this paper is to study adaptive mesh techniques, using a posteriori error estimates, for the finite element solution of the Navier-Stokes equations modeling steady and unsteady flows of an incompressible viscous fluid. Among existing operator splitting techniques, the θ-scheme is used for time integration of the Navier-Stokes equations. Then, a posteriori error estimates, based on the solution of a local system for each triangular element, are presented in the framework of the generalized incompressible Stokes problem, followed by its practical application to the case of incompressible Navier-Stokes problem. Hierarchical mesh adaptive techniques are developed in response to the a posteriori error estimation. Numerical simulations of viscous flows associated with selected geometries are performed and discussed to demonstrate the accuracy and efficiency of our methodology.     

Improved cache utilization and preconditioner efficiency through use of a space-filling curve mesh element- and vertex-reordering technique

Solving partial differential equations using finite element (FE) methods for unstructured meshes that contain billions of elements is computationally a very challenging task. While parallel implementations can deliver a solution in a reasonable amount of time, they suffer from low cache utilization due to unstructured data access patterns. In this work, we reorder the way the mesh vertices and elements are stored in memory using Hilbert space-filling curves to improve cache utilization in FE methods for unstructured meshes. This reordering technique enumerates the mesh elements such that parallel threads access shared vertices at different time intervals, reducing the time wasted waiting to acquire locks guarding atomic regions. Further, when the linear system resulting from the FE analysis is solved using the preconditioned conjugate gradient method, the performance of the block-Jacobi preconditioner also improves, as more nonzeros are present near the stiffness matrix diagonal. Our results show that our reordering reduces the L1 and L2 cache miss-rates in the stiffness matrix assembly step by about 50 and 10 %, respectively, on a single-core processor. We also reduce the number of iterations required to solve the linear system by about 5 %. Overall, our reordering reduces the time to assemble the stiffness matrix and to solve the linear system on a 4-socket, 48-core multi-processor by about 20 %.     

Backward difference replacements of the space derivative in first-order hyperbolic equations

Two families of two-time level difference schemes are developed for the numerical solution of first-order hyperbolic partial differential equations with one space variable. The space derivative is replaced by (i) a first-order, (ii) a second-order backward difference approximant, and the resulting system of first-order ordinary differential equations is solved using A₀-stable and L₀-stable methods. The methods are used explicitly and are inexpensive to implement.The methods are tested on a number of problems from the literature involving wave-form solutions, increasing solutions with discontinuities in function values or first derivatives across a characteristic, and exponentially decaying solutions.     

基于LS-SVM方法求高阶线性ODE近似解

对于线性常微分方程,解析解方便定性分析和实际应用,然而大多数微分方程没有解析解。回归的方法被应用获取近似解析解,其中最小二乘支持向量机（LS-SVM）是目前为止最好的方法。但是该方法不仅需要对核函数求高阶导数而且需要求解一个大的线性方程组。为此,把高阶线性常微分方程转化为一阶线性常微分方程组,构建含有一阶导数形式的LS-SVM回归模型。该模型利用最小化误差函数去获得合适的参数,最终通过求解三个小的线性方程组获得高精度的近似解（连续、可微）。实验结果验证了该方法的有效性。     

Piecewise-linearized and linearized θ-methods for ordinary and partial differential equations

Initial- and boundary-value problems appear frequently in many branches of physics. In this paper, several numerical methods, based on linearization techniques, for solving these problems are reviewed. First, piecewise-linearized methods and linearized θ-methods are considered for the solution of initial-value problems in ordinary differential equations. Second, piecewise-linearized techniques for two-point boundary-value problems in ordinary differential equations are developed and used in conjunction with a shooting method. In order to overcome the lack of convergence associated with shooting, piecewise-linearized methods which provide piecewise analytical solutions and yield nonstandard finite difference schemes are presented. Third, methods of lines in either space or time for the solution of one-dimensional convection-reaction-diffusion problems that transform the original problem into an initial- or boundary-value one are reviewed. Methods of lines in time that result in boundary-value problems at each time step can be solved by means of the techniques described here, whereas methods of lines in space that yield initial-value problems and employ either piecewise-linearized techniques or linearized θ-methods in time are also developed. Finally, for multidimensional problems, approximate factorization methods are first used to transform the multidimensional problem into a sequence of one-dimensional ones which are then solved by means of the linearized and piecewise-linearized methods presented here.     

Similarity solutions for axisymmetric plane radial power law fluid flows through a porous medium

This paper investigates the unsteady flow of non-Newtonian fluids of power low behavior through a porous medium in a plane radial geometry. The equation governing the flow is a nonlinear parabolic partial differential equation with a source term whose solution satisfies certain fixed and moving boundary conditions. The attention is focused on the finding of similarity solution when the fixed boundary condition and the source term satisfy certain restrictions. In this case similarity transformations are determined and the resulting ordinary differential equations are deduced. For shear thinning fluids the existence of a pressure disturbance front moving with finite velocity is shown and expression for its location as a function of time is determined. The solutions in closed form have been given for certain particular cases where the resulting differential equations can be analytically solved. A numerical procedure has also been presented.     

Modelling telescopic boom--The plane case: part I

We shall briefly present an idea for the modelling flexible telescopic boom using a non-linear finite element method. The boom is assembled by Reissner’s geometrically exact beam elements. The sliding boom parts are coupled together by the element, where a slide-spring is coupled to beam with the aid of a master-slave technique. This technique yields system equations without algebraic constraints. Telescopic movement is achieved by the rod element with varying length and the connector element expressing the chains. The structural dynamic calculation model is converted to first order ordinary differential equations by adding nodal velocities to state variable, which is solved by the Rosenbrock-W integration method.     

The standard H∞ problem in systems with a singleinput delay

The standard H_∞ problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration     

$$L_2$$ Stability of Explicit Runge–Kutta Schemes

Explicit Runge–Kutta methods are standard tools in the numerical solution of ordinary differential equations (ODEs). Applying the method of lines to partial differential equations, spatial semidiscretisations result in large systems of ODEs that are solved subsequently. However, stability investigations of high-order methods for transport equations are often conducted only in the semidiscrete setting. Here, strong-stability of semidiscretisations for linear transport equations, resulting in ODEs with semibounded operators, are investigated. For the first time, it is proved that the fourth-order, ten-stage SSP method of Ketcheson (SIAM J Sci Comput 30(4):2113–2136, 2008) is strongly stable for general semibounded operators. Additionally, insights into fourth-order methods with fewer stages are presented.     

An asymptotic numerical method for fourth order singular perturbation problems with a discontinuous source term

Singularly perturbed two-point boundary-value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter ? multiplying the highest derivative, and suitable boundary conditions. In this paper a computational method for solving this system is presented. In this method we first find the zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the numerical method, which is constructed for this problem and which involves an appropriate piecewise-uniform mesh.     

Numerical solution of singular ODE eigenvalue problems in electronic structure computations

We put forward a new method for the solution of eigenvalue problems for (systems of) ordinary differential equations, where our main focus is on eigenvalue problems for singular Schrödinger equations arising for example in electronic structure computations. In most established standard methods, the generation of the starting values for the computation of eigenvalues of higher index is a critical issue. Our approach comprises two stages: First we generate rough approximations by a matrix method, which yields several eigenvalues and associated eigenfunctions simultaneously, albeit with moderate accuracy. In a second stage, these approximations are used as starting values for a collocation method which yields approximations of high accuracy efficiently due to an adaptive mesh selection strategy, and additionally provides reliable error estimates. We successfully apply our method to the solution of the quantum mechanical Kepler, Yukawa and the coupled ODE Stark problems.     

Feedback control of linear systems with distributed delay

The problem of optimal control of linear systems containing lumped delay, given by differential-difference equations, has been pursued by several authors. However, transportation-lags are better described by distributed delays giving systems that are described by a set of coupled partial and ordinary differential equations. The lumped part of the system is described by ordinary differential equations and the distributed part of the system is described by partial differential equations. The lumped as well as distributed parts are subject to control. The present paper discusses the control of such systems with quadratic performance measures. Riccati-like equations are derived and a technique for their numerical solution is presented.