Initial Value Partial Differential Equations Research Articles

A time-spectral method for initial-value problems using a novel spatial subdomain scheme

We analyse a novel subdomain scheme for time-spectral solution of initial-value partial differential equations. In numerical modelling spectral methods are commonplace for spatially dependent syste...

A Spectral Method in Time for Initial-Value Problems

A time-spectral method for solution of initial value partial differential equations is outlined. Multivariate Chebyshev series are used to represent all temporal, spatial and physical parameter domains in this generalized weighted residual method (GWRM). The approximate solutions obtained are thus analytical, finite order multivariate polynomials. The method avoids time step limitations. To determine the spectral coefficients, a system of algebraic equations is solved iteratively. A root solver, with excellent global convergence properties, has been developed. Accuracy and efficiency are controlled by the number of included Chebyshev modes and by use of temporal and spatial subdomains. As examples of advanced application, stability problems within ideal and resistive magnetohydrodynamics (MHD) are solved. To introduce the method, solutions to a stiff ordinary differential equation are demonstrated and discussed. Subsequently, the GWRM is applied to the Burger and forced wave equations. Comparisons with the explicit Lax-Wendroff and implicit Crank-Nicolson finite difference methods show that the method is accurate and efficient. Thus the method shows potential for advanced initial value problems in fluid mechanics and MHD.

Dynamic response and reliability analysis of structures with uncertain parameters

AbstractAn original approach for dynamic response and reliability analysis of stochastic structures is proposed. The probability density evolution equation is established which implies that incremental rate of the probability density function is related to the structural response velocity. Therefore, the response analysis of stochastic structures becomes an initial‐value partial differential equation problem. For the dynamic reliability problem, the solution can be derived through solving the probability density evolution equation with an initial value condition and an absorbing boundary condition corresponding to specified failure criterion. The numerical algorithm for the proposed method is suggested by combining the precise time integration method and the finite difference method with TVD scheme. To verify and validate the proposed method, a SDOF system and an 8‐storey frame with random parameters are investigated in detail. In the SDOF system, the response obtained by the proposed method is compared with the counterparts by the exact solution. The responses and the reliabilities of a frame with random stiffness, subject to deterministic excitation or random excitation, are evaluated by the proposed method as well. The mean, the standard deviation and the reliabilities are compared, respectively, with the Monte Carlo simulation. The numerical examples verify that the proposed method is of high accuracy and efficiency. Moreover, it is found that the probability transition of structural responses is like water flowing in a river with many whirlpools, showing complexity of probability transition process of the stochastic dynamic responses. Copyright © 2004 John Wiley & Sons, Ltd.

Transport and diffusion of material quantities on propagating interfaces via level set methods

We develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework. Level set methods are computational techniques for tracking moving interfaces; they work by embedding the propagating interface as the zero level set of a higher dimensional function, and then approximate the solution of the resulting initial value partial differential equation using upwind finite difference schemes. The traditional level set method works in the trace space of the evolving interface, and hence disregards any parameterization in the interface description. Consequently, material quantities on the interface which themselves are transported under the interface motion are not easily handled in this framework. We develop model equations and algorithmic techniques to extend the level set method to include these problems. We demonstrate the accuracy of our approach through a series of test examples and convergence studies.

LEVEL SET METHODS FOR FLUID INTERFACES

▪ Abstract We provide an overview of level set methods, introduced by Osher and Sethian, for computing the solution to fluid-interface problems. These are computational techniques that rely on an implicit formulation of the interface, represented through a time-dependent initial-value partial-differential equation. We discuss the essential ideas behind the techniques, the coupling of these techniques to finite-difference methods for incompressible and compressible flow, and a collection of applications including two-phase flow, ship hydrodynamics, and ink-jet-printhead design.

A fast marching level set method for monotonically advancing fronts.

A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial differential equation for a propagating level set function and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. This paper describes a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations, and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithographic development calculations in microchip manufacturing, and arrival time problems in control theory.

Theory, algorithms, and applications of level set methods for propagating interfaces

We review recent work on level set methods for following the evolution of complex interfaces. These techniques are based on solving initial value partial differential equations for level set functions, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The methodology results in robust, accurate, and efficient numerical algorithms for propagating interfaces in highly complex settings. We review the basic theory and approximations, describe a hierarchy of fast methods, including an extremely fast marching level set scheme for monotonically advancing fronts, based on a stationary formulation of the problem, and discuss extensions to multiple interfaces and triple points. Finally, we demonstrate the technique applied to a series of examples from geometry, material science and computer vision, including mean curvature flow, minimal surfaces, grid generation, fluid mechanics, combustion, image processing, computer vision, and etching, deposition and lithography in the microfabrication of electronic components.

Ballooning instabilities in tokamaks with sheared toroidal flows

The ballooning-mode eikonal representation is applied to the linearized incompressible magnetohydrodynamic (MHD) equations in axisymmetric systems with toroidal mass flows to obtain a set of initial value partial differential equations in which the time t and the poloidal angle theta are the independent variables. To derive these equations, the eikonal function S is assumed to satisfy the usual condition B.gradS=0 to guarantee that the modes vary slowly along the magnetic field. In addition, to resolve the V.grad operator acting on perturbed quantities, the eikonal must also satisfy the condition dS/dt=0. this induces a Doppler shift in S. This description of the instability, however, is incompatible with normal mode solutions of the MHD equations because the wave vector gradS becomes time dependent when the velocity shear is finite. Nevertheless, the author is able to investigate the effects of the sheared toroidal flows on localized ballooning instabilities because the initial value formulation of the problem developed does not constrain the solutions to evolve as exp (i omega t). Fixed boundary MHD equilibria with isothermal toroidal flows that model the JET device are generated numerically with a variational inverse moments code. As the initial value evaluations are evolved in time, periodic burst of ballooning activity are observed which are correlated with the formation of a ballooning structure at the outside edge of the torus that becomes displaced by 2 pi in the extended poloidal angle domain from one burst to the next. The velocity shear has a stabilizing influence on plasma ballooning.

Correction of atmospheric effects on earth imaging (extended inverse method)

The usual problem of correction of atmospheric scattering effects involves the inverse process of a nonlinear partial differential equation in three-dimensional space. The method developed here involves solving three initial value partial differential equations. Two of them are Chandrasekhar's type one- dimensional radiative transfer equations. The third equation is a linear partial differential equation in three-dimensional space with initial value.

Product formulas for solutions of initial value partial differential equations, II

Under consideration are semi-groups generated by certain partial differential operators whose coefficients are polynomials. Product formulas are developed for these semi-groups, involving integral operators and the fundamental solution of the one-dimensional parabolic equation ∂u ∂t =( (−1) n 2−1 n! ( ∂ uu ∂x n ) . These formulas reveal special support properties of the semi-group when the generator is degenerate.

Product formulas for solutions of initial value partial differential equations, I

Families of operators which approximate semi-groups or evolution systems generated by partial differential operators are constructed. Product formulas are used to recover these semi-groups or evolution systems through product integrals. Conditions on generators are provided under which its semi-group or evolution system can be approximated in this way by families of specific types of operators.

Variational design of finite-difference schemes for initial value problems with an integral invariant

A technique for deriving a finite-difference scheme to solve initial value partial-differential equations is presented. The set of partial-differential equations is assumed to possess one or more invariant integral quantities. In fluid dynamics, the integral of the total energy over the domain is frequently assumed invariant. This technique is based on the variational method and constrains the finite-difference scheme to satisfy the conservation law(s). The technique is discussed by considering, as an example, a set of linear, shallow-water equations on a rotating plane, and extending it to a nonlinear case.

Formulation of stable difference schemes for systems of initial-value partial differential equations

A general system of initial-value partial differential equations is classified into four categories based on the partial differential operators which define the equations. Specific combinations of the operators are termed “invariants” since they are common to all finite difference approximations to the system of equations. The “invariants” are used to a priori determine if one may formulate a stable difference approximation to a system of partial differential equations. This is in essence a numerical existence theory.

Modal simulation of a distributed parameter system

This paper presents an analog/hybrid simulation method for linear and nonlinear distributed para meter systems. The modal method is based on a functional approximation for the computer solution of initial value partial differential equation prob lems using eigenfunctions. First, the mathematical foundations are reviewed; then the questions concerning the technical realiza tion on an analog/hybrid computer are treated. The simulation results for two examples of chemical processes are discussed.

R70-6 Use of Functional Approximation Methods in the Computer Solution of Initial Value Partial Differential Equation Problems

The purpose of this paper is to provide a broad review of a class of unconventional computational methods for solving certain partial differential equations. The most widely used analog and digital techniques for solving initial-value problems involve the discretization of the space domain, with the aid of finite difference approximations. In the case of digital methods, there results a set of algebraic equations which must be solved for each step in the time domain. In analog techniques, finite difference approximations lead to a set of simultaneous ordinary differential equations, one for each grid point in this space domain. These equations can then be integrated using analog integrators. Although quite general in their applicability, finite differences methods often entail excessive and uneconomical computational efforts. This shortcoming, as well as the desire to explore new potential applications for hybrid computers, has led a number of investigators to experiment with alternative techniques, techniques which the author of the paper under review has chosen to classify as functional approximation methods.

On stability of numerical methods for systems of initial-value partial differential equations

On stability of numerical methods for systems of initial-value partial differential equations

Use of Functional Approximation Methods in the Computer Solution of Initial Value Partial Differential Equation Problems

Methods of functional approximation for the computer solution of initial value partial differential equation problems provide a device by which these solutions can be approximated by those of initial value problems in sets of ordinary differential equations. A number of ways to achieve this have been suggested, some of them general, some of them utilizing specific properties of the equations at hand. What all these methods have in common is the fact that the solution u(x, t) of a partial differential equation in space x and time t is approximated by a function u*(a1(t), a2(t), . . ., aN(t)), where the dependence upon x is prescribed. In most applications, u* is linear in the ai(t), i.e., u*= =ai(t)fi(x). The ai(t) satisfy a set of ordinary differential equations obtained as the result of the approximation process. This system of ordinary differential equations is then integrated by classical analog or digital computer methods.