Comparison of continuous and discontinuous finite element methods for parabolic differential equations employing implicit time integration

This article compares the computational cost, stability, and accuracy of continuous and discontinuous Galerkin Finite Element Methods (GFEM) for various parabolic differential equations including the advection–diffusion equation, viscous Burgers’ equation, and Turing pattern formation equation system. The results show that, for implicit time integration, the continuous GFEM is typically 5–20 times less computationally expensive than the discontinuous GFEM using the same finite element mesh and element order. However, the discontinuous GFEM is significantly more stable than the continuous GFEM for advection dominated problems and is able to obtain accurate approximate solutions for cases where the classic, un-stabilized continuous GFEM fails.     

Comparative study of six explicit and two implicit finite difference schemes for solving one-dimensional parabolic partial differential equations

Eight finite difference schemes used in solving parabolic partial differential equations are compared with respect to accuracy, execution time and programming effort. The analysis presented is useful in selecting the appropriate numerical scheme depending on the emphasis placed upon accuracy, execution time or programming effort.     

An algorithm for computing a companion block diagonal form for a system of linear differential equations

In this paper we describe a new algorithm which reduces in a finite number of steps a linear system of differential equations to a companion block diagonal form. This form is particularly convenient if one wishes to compute invariants at singularities.     

An efficient method of solving the Navier–Stokes equations for flow control

A new method of solving the Navier–Stokes equations efficiently by reducing their number of modes is proposed in the present paper. It is based on the Karhunen–Loève decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena, and consequently reduce the Navier–Stokes equation defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom. The present algorithm is well suited for the problems of flow control or optimization, where one has to compute the flow field repeatedly using the Navier–Stokes equation but one can also estimate the approximate solution space of the flow field based on the range of control variables. The low-dimensional dynamic model of viscous fluid flow derived by the present method is shown to produce accurate flow fields at a drastically reduced computational cost when compared with the finite difference solution of the Navier–Stokes equation. © 1998 John Wiley & Sons, Ltd.     

The chaotic behaviour of piecewise smooth differential equations on two-dimensional torus and sphere

This paper studies the global dynamics of piecewise smooth differential equations defined in the two-dimensional torus and sphere in the case when the switching manifold breaks the manifold into two connected components. Over the switching manifold, we consider the Filippov's convention for discontinuous differential equations. The study of piecewise smooth dynamical systems over torus and sphere is common for maps and up to where we know this is the first characterization for piecewise smooth flows arising from solutions of differential equations. We provide conditions under generic families of piecewise smooth equations to get periodic and dense trajectories. Considering these generic families of piecewise differential equations, we prove that a non-deterministic chaotic behaviour appears. Global bifurcations are also classified.     

A generalized Ritz method for partial differential equations in domains of arbitrary geometry using global shape functions

A boundary element method (BEM)-based variational method is presented for the solution of elliptic PDEs describing the mechanical response of general inhomogeneous anisotropic bodies of arbitrary geometry. The equations, which in general have variable coefficients, may be linear or nonlinear. Using the concept of the analog equation of Katsikadelis the original equation is converted into a linear membrane (Poisson) or a linear plate (biharmonic) equation, depending on the order of the PDE under a fictitious load, which is approximated with radial basis function series of multiquadric (MQ) type. The integral representation of the solution of the substitute equation yields shape functions, which are global and satisfy both essential and natural boundary conditions, hence the name generalized Ritz method. The minimization of the functional that produces the PDE as the associated Euler–Lagrange equation yields not only the Ritz coefficients but also permits the evaluation of optimal values for the shape parameters of the MQs as well as optimal position of their centers, minimizing thus the error. If a functional does not exists or cannot be constructed as it is the usual case of nonlinear PDEs, the Galerkin principle can be applied. Since the arising domain integrals are converted into boundary line integrals, the method is boundary-only and, therefore, it maintains all the advantages of the pure BEM. Example problems are studied, which illustrate the method and demonstrate its efficiency and great accuracy.     

An analytical-numerical method for fast evaluation of probability densities for transient solutions of nonlinear Itô’s stochastic differential equations

Probability densities for solutions of nonlinear Itô’s stochastic differential equations are described by the corresponding Kolmogorov-forward/Fokker-Planck equations. The densities provide the most complete information on the related probability distributions. This is an advantage crucial in many applications such as modelling floating structures under the stochastic-load due to wind or sea waves. Practical methods for numerical solution of the probability density equations are combined, analytical-numerical techniques. The present work develops a new analytical-numerical approach, the successive-transition (ST) method, which is a version of the path-integration (PI) method. The ST technique is based on an analytical approximation for the transition probability density. It enables the PI approach to explicitly allow for the damping matrix in the approximation. This is achieved by extending another method, introduced earlier for bistable nonlinear reaction-diffusion equations, to the probability density equations. The ST method also includes a control for the size of the time-step. The overall accuracy of the ST method can be tested on various nonlinear examples. One such example is proposed. It is one-dimensional nonlinear Itô’s equation describing the velocity of a ship maneuvering along a straight line under the action of the stochastic drag due to wind or sea waves. Another problem in marine engineering, the rolling of a ship up to its possible capsizing is also discussed in connection with the complicated damping matrix picture. The work suggests a few directions for future research.     

A novel method for integrating first‐ and second‐order differential equations in elastohydrodynamic lubrication for the solution of smooth isothermal,line contact problems

This paper contains details of recent developments in the analysis of elastohydrodynamic lubrication problems using the finite element method. A steady state isothermal finite element formulation of the smooth line contact problem with Newtonian lubricant behaviour is presented containing both first‐ and second‐order formulations of the hydrodynamic equation. Previous problems with the limited range of applicability of both first‐ and second‐order finite difference solutions have been overcome by summing both the first‐ and second‐order equations' weighted contributions. Application of the method to a range of problems unattainable by either single first‐ or second‐order formulations is presented. Copyright © 1999 John Wiley & Sons, Ltd.