Mixed symbolic–numerical computations with general DAEs I: System properties

Differential-algebraic equations (DAEs) arise in many ways in many types of problems. In this expository paper we discuss a variety of situations where we have found mixed symbolic-numerical calculations to be essential. The paper is designed to both familiarize the reader with several fundamental DAE ideas and to present some applications. The situations discussed include the analysis of DAEs, the solution of DAEs, and applications which include DAEs. Both successes and challenges will be presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Mixed symbolic–numerical computations with general DAEs II: An applications case study

A variety of theorems and properties of nonlinear DAEs were discussed in part I. This paper illustrates many of these ideas within the context of analyzing a specific nonlinear system that exhibits a variety of interesting features. This revised version was published online in June 2006 with corrections to the Cover Date.     

A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems

In this article, we develop an explicit symmetric linear phase-fitted four-step method with a free coefficient as parameter. The parameter is used for the optimization of the method in order to solve efficiently the Schrödinger equation and related oscillatory problems. We evaluate the local truncation error and the interval of periodicity as functions of the parameter. We reveal a direct relationship between the periodicity interval and the local truncation error. We also measure the efficiency of the new method for a wide range of possible values of the parameter and compare it to other well known methods from the literature. The analysis and the numerical results help us to determine the optimal values of the parameter, which render the new method highly efficient.     

A parallel computational scheme with ninth-order multioperator approximations and its application to direct numerical simulation

The properties of ninth-order multioperator compact schemes based on known third-and fifth-order compact approximations are examined. The domains where the multioperators have fixed signs are determined numerically. The numerical results are compared with the exact solution to the Burgers equation. The multioperator schemes are applied to the problem of vortex sheet roll-up.     

A numerical scheme for multi-point special boundary-value problems and application to fluid flow through porous layers

In this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi-point boundary-value problems over adjacent domains. In each subdomain the solution is governed by a different equation. The solutions are required to be smooth across the interface nodes. The approach is based on using finite difference approximation of the derivatives at the interface nodes. Smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations. A modified multi-dimensional Newton’s method is proposed for solving the nonlinear system. The accuracy of the proposed scheme is validated by examples whose exact solutions are known. The proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media.     

A rational generalization of Fan’s method and its application to generalized shallow water wave equations

In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.     

A improved F‐expansion method and its application to the Zhiber–Shabat equation

Based on the F‐expansion method and Exp‐function method, an improved F‐expansion method is introduced. As illustrative examples, the exact solutions expressed by exponential function, hyperbolic functions, logarithmic function, and other type of functions for the Zhiber–Shabat equation are derived. Some previous results are extended. Copyright © 2012 John Wiley & Sons, Ltd.     

A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow

Biharmonic equations have many applications, especially in fluid and solid mechanics, but is difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary conditions. The irregular domain is embedded into a rectangular region and the biharmonic equation is decoupled to two Poisson equations. An auxiliary unknown quantity Δu along the boundary is introduced so that fast Poisson solvers on irregular domains can be used. Non-trivial numerical examples show the efficiency of the proposed method. The number of iterations of the method is independent of the mesh size. Another key to the method is a new interpolation scheme to evaluate the residual of the Schur complement system. The new biharmonic solver has been applied to solve the incompressible Stokes flow on an irregular domain.