General Initial Value Problems Research Articles

Similarity solutions for a class of hyperbolic integrodifferential equations

For a class of hyperbolic partial integrodifferential equations of the form $u_{tt} - u_{xx} + a*u_{xx} = 0$, fundamental solutions are found that depend on the similarity variable $\xi =x(t-|x|)^{-\alpha}$, where $\alpha \in (0,1)$ and the integral kernel $a$ behaves like $t^{-\alpha}$ near $t=0$. The asymptotic behavior of these solutions in various scaling limits and their regularity is discussed. Applications to solutions of general initial-value problems of such equations are given.

The initial-value problem for the Kelvin-Helmholtz instabilities of high-velocity and magnetized shear layers

The general initial-value problem for the linear Kelvin-Helmholtz instability of arbitrarily compressible velocity shear layers is considered for both the unmagnetized and magnetized cases. The time evolution of the physical quantities characterizing the layer is treated using Laplace transform techniques. Singularity analysis of the resulting equations using Fuchs-Frobenius theory yields the large-time asymptotic solutions. The instability is found to remain, within the linear theory, of the translationally convective or shear type. No onset of rotational or vortex motion, i.e., formation of “coherent structures” occurs.

Application of the fourth-order Runge-Kutta method for the solution of high-order general initial value problems

A Runge-Kutta algorithm for nth order initial value problem (IVP), x ( n) ( t) = f( t, x, x′,…, x ( n−1) , t > t 0, with x( t 0), x ( r) ( t 0) given ( r = 1,2,…, n −1), is developed by means of the algorithm used for the first-order IVP. As an example, the method is tested on a fourth-order IVP involving nonlinear terms and on a linear fourth-order problem from ship dynamics.

The general and asymptotic solution of the elastic wave equation in an unbounded medium

New forms and new properties are given for the general solution of the three-dimensional dynamic equation for the displacement vector in an isotropic linear elastic solid. The general initial value problem in an unbounded medium with no sources is solved. The general solution at an arbitrary time is given in terms of two asymptotic vector fields. One is thereby able to obtain the complete and exact solution, including initial conditions, which corresponds to a specified or desired asymptotic solution, valid for large times in the future. The complete elastic field energy of an arbitrary exact solution is also shown to depend, in a simple way, only on the two asymptotic vector fields.

A modified Adams method for nonstiff and mildly stiff initial value problems

Adams predictor-corrector methods, and explicit Runge–Kutta formulas, have been widely used for the numerical solution of nonstiff initial value problems. Both of these classes of methods have certain drawbacks, however, and it has long been the aim of numerical analysts to derive a class of formulas that has the advantages of both Adams and Runge–Kutta methods and the disadvantages of neither! In this paper we derive a class of modified Adams formulas that attempts to achieve this aim. When used in a certain precisely defined predictor-corrector mode, these new formulas require three function evaluations per step, but have much better stability than Adams formulas. This improved stability makes the modified Adams formulas particularly effective for mildly stiff problems, and some numerical evidence of this is given. We also consider the performance of the new class of methods on the well-known DETEST test set to show their potential on general nonstiff initial value problems.

An exact Riemann solver for detonation products

Numerical methods based upon the Riemann Problem are considered for solving the general initial-value problem for the Euler equations applied to real gases. Most of such methods use an approximate solution of the Riemann problem when real gases are involved. These approximate Riemann solvers do not yield always a good resolution of the flow field, especially for contact surfaces and expansion waves. Moreover, approximate Riemann solvers cannot produce exact solutions for the boundary points. In order to overcome these shortcomings, an exact solution of the Riemann problem is developed, valid for real gases. The method is applied to detonation products obeying a 5th order virial equation of state, in the shock-tube test case. Comparisons between our solver, as implemented in Random Choice Method, and finite difference methods, which do not employ a Riemann solver, are given.

Riemann problems and the WAF method for solving the two-dimensional shallow water equations

An exact Riemann solver for the shallow water equations along with several approximate Riemann solvers are presented. These solutions are then used locally to help compute numerically the global solution of the general initial boundary value problem for the shallow water equations. The numerical method used is the weighted average flux method (WAF) proposed by the author. This is a conservative, shock capturing high resolution TVD method. For shallow water flows where nonlinear effects are important or where abrupt changes (hydraulic jumps) are to be expected the present algorithms can be useful in practice. One and two-dimensional solutions are presented to assess both the Riemann solvers and the WAF method.

A linearized Riemann solver for the time-dependent Euler equations of gas dynamics

The time-dependent Euler equations of gas dynamics are a set of nonlinear hyperbolic conservation laws that admit discontinuous solutions (e. g. shocks). In this paper we are concerned with Riemann-problem-based numerical methods for solving the general initial-value problem for these equations. We present an approximate, linearized Riemann solver for the time-dependent Euler equations. The solution is direct and involves few, and simple, arithmetic operations. The Riemann solver is then used, locally, in conjunction with the weighted average flux numerical method to solve the time-dependent Euler equations in one and two space dimensions with general initial data. For flows with shock waves of moderate strength the computed results are very accurate. For severe flow régimes we advocate the use of the present linearized Riemann solver in combination with the exact Riemann solver in an adaptive fashion. Numerical experiments demonstrate that such an approach can be very successful. One-dimensional and two-dimensional test problems show that the linearized Riemann solver is used in over 99% of the flow field producing net computing savings by a factor of about 2. A reliable and simple switching criterion is also presented. Results show that the adaptive approach effectively provides the resolution and robustness of the exact Riemann solver at the computing cost of the simple linearized Riemann solver. The relevance of the present methods concerns the numerical solution of multi-dimensional problems accurately and economically.

Diffusing-vortex numerical scheme for solving incompressible navier-stokes equations

A new numerical algorithm, the diffusing-vortex method for time-dependent two-dimensional Navier-Stokes equations, which was previously presented and applied to the incompressible viscous flow past a circular cylinder with high Reynolds number by Lu and Shen [1], is further developed for extension to general two-dimensional initial value problems and boundary value problems. The new algorithm consists of two time steps is a simulation cycle: a Lagrangian convection simulation for the first time step and a diffusion simulation through the use of new vortex points at dixed Eulerian mesh points for the second time step. The mathematical mechanisms of computation behind this algorithm, and its characteristics of convergence and accuracy, are analyzed in applications for the following problems: (1) an initial value problem involving the decay of a single vortex of finite size and the decay and interaction of a vortex-pair of two finite-core regions; (2) a boundary value problem: the unsteady flow field around a rotating cylinder with high Reynolds number up to Re d = U · D/ v < 10,000. Numerical results are compared with either exact solutions or other numerical methods. The numerical advantages of the diffusing-vortex scheme over other conventional vortex methods, Cloud-in-Cell methods, particle methods, and some finite difference schemes are evaluated in terms of reducing total CPU time, avoiding cutoff procedures, and sidestepping various interpolations.

Chaos in a model of forced quasi-geostrophic flow over topography: an application of Melnikov's method

We demonstrate the existence of a chaotic invariant set of solutions of an idealized model for wind-forced quasi-geostrophic flow over a continental margin with variable topography. The model (originally formulated to investigate mean flow generation by topographic wave drag) has bottom topography that slopes linearly offshore and varies sinusoidally alongshore. The alongshore topographic scales are taken to be short compared to the cross-shelf scale, allowing Hart's (1979) quasi-two-dimensional approximation, and the governing equations reduce to a non-autonomous system of three coupled nonlinear ordinary differential equations. For weak (constant plus time-periodic) forcing and weak friction, we apply a recent extension (Wiggins &amp; Holmes 1987) of the method of Melnikov (1963) to test for the existence of transverse homoclinic orbits in the model. The inviscid unforced equations have two constants of motion, corresponding to energy E and enstrophy M, and reduce to a one-degree-of-freedom Hamiltonian system which, for a range of values of the constant G = E − M, has a pair of homoclinic orbits to a hyperbolic saddle point. Weak forcing and friction cause slow variations in G, but for a range of parameter values one saddle point is shown to persist as a hyperbolic periodic orbit and Melnikov's method may be applied to study the perturbations of the associated homoclinic orbits. In the absence of time-periodic forcing, the hyperbolic periodic orbit reduces to the unstable fixed point that occurs with steady forcing and friction. The method yields analytical expressions for the parameter values for which sets of chaotic solutions exist for sufficiently weak time-dependent forcing and friction. The predictions of the perturbation analysis are verified numerically with computations of Poincaré sections for solutions in the stable and unstable manifolds of the hyperbolic periodic orbit and with computations of solutions for general initial-value problems. In the presence of constant positive wind stress τ0 (equatorward on eastern ocean boundaries), chaotic solutions exist when the ratio of the oscillatory wind stress τ1 to the bottom friction parameter r is above a critical value that depends on τ0/r and the bottom topographic height. The analysis complements a previous study of this model (Samelson &amp; Allen 1987), in which chaotic solutions were observed numerically for weak near-resonant forcing and weak friction.

A modified numerov integration method for general second order initial value Problems

We present an implicit eighth-order finite difference method for the general second order non-linear differential equation Yn=f(t,y,y) subject to the initial conditions y(to) =yo, y(to)=yO. The method is based on mixed interpolation containing a parameter which can be used to improve the accuracy. The same technique gives rise to an explicit sixth-order integration scheme. Numerical solutions of problems are given to illustrate both methods.

On a Moving Boundary Solution to the Fokker-Planck Equation for Particle Transport in Turbulent Flows with Absorbing Boundaries

A Fokker—Planck equation describing the transport of particles in turbulent flows is considered. The initial value problem with perfectly absorbing boundary conditions on the wall is solved by introducing a characteristic line in phase space. It is found that the solution domain in phase space decomposes into three regions which are connected by two moving boundaries, and the moving boundaries can be found explicitly. For Gaussian-type initial conditions, the solution of the Fokker—Planck equation can be obtained in each of the three regions by solving a diffusion equation. An approximation procedure for the general initial value problem is established and the approximate solution sequence is shown to be convergent in a certain Hilbert space.

Spectral Methods for Initial Boundary Value Problems—An Alternative Approach

This paper presents a new approach to spectral methods for initial boundary value problems. A filtered version of the partial differential equation and the initial and boundary conditions at an overdetermined set of points are collocated. As an approximate solution, the function is chosen that belongs to an appropriate finite-dimensional space and minimizes a weighted average of the residuals at these points. It is proved that the approximate solution converges to the actual solution at a spectral rate of accuracy in both space and time. The proof is based on a priori energy estimates that have been proved for such systems. Although this method is restricted here to hyperbolic initial boundary value problems and Chebyshev polynomials, it generalizes to general initial boundary value problems, boundary value problems, and Gegenbauer polynomials.

Global extrapolation of numerical methods for initial value problems

Two-, three- and four-grid extrapolation procedures are considered for the special initial value problem of arbitrary order q > 1. A number of new extrapolation formulas are developed for convergent numerical methods of arbitrary order p≧2. The global extrapolations of three existing P-stable numerical methods are considered and tested on two second order initial value problems from the literature. Global extrapolation procedures for general initial value problems are also considered.

Bounded and unbounded solutions of the von Kármán vortex trail

The general initial value problem for the linearized von Karman vortex trail is solved. In particular, the fundamental differences between von Karman’s normal mode solutions and aperiodic solutions with compact support are discussed. Within the natural space ℓ 2 {\ell _2} , it is shown that the von Karman trail at its special aspect ratio (neutrally stable case) supports unbounded solutions. Two invariants of the equations determine whether a solution is bounded or unbounded. The asymptotic behaviour of aperiodic solutions is discussed. It is found that unbounded solutions grow at the rate O ( l n τ ) O\left ( {ln \tau } \right ) in ℓ ∞ {\ell _\infty } while bounded solutions decay as O ( 1 / τ ) O\left ( {1/\sqrt \tau } \right ) . In addition, a Loschmidt’s demon is constructed for the reversible dispersive equations which demonstrates the focussing effect in the von Karman trail.

A modification of the stiefel-bettis method for nonlinearly damped oscillators

We report a modification of the Stiefel-Bettis method which is of trigonometric order one and of polynomial order two for the general second order initial value problems. We also discuss the modified Stiefel-Bettis method made explicit for the undamped nonlinear oscillators. Numerical solution of problems are given to illustrate the methods.

A family of numerical methods for the solution of high-order general initial value problems

A family of numerical methods is developed for the solution of the general initial value problem y ( N) ( t) = f( t, y, y t , …, y ( N−1) ), t> t 0, with y( t 0, y ( r) ( t 0) given ( r = 1, 2, …, N−1). The orders of the methods are seen to be one or two and global extrapolation is used to increase the order of a given method by one or two powers. The methods are tested on the Blasius nonlinear third-order initial value problem and on a linear fourth-order problem from ship dynamics.

Exact solutions for wave equations of two-layered media with smooth transition

The wave equation c2(x)uxx−utt=0 is solved for wave speeds c(x) corresponding to two-layered media with smooth transition from layer to layer. The wave speed c(x) has four free parameters to fit a given medium. Solutions are constructed from invariant solutions of a related system of first-order partial differential equations that admit a four-parameter symmetry group. These solutions are superposed to solve general initial value problems for data with compact support; the computation of the superposition coefficients uses elementary Fourier analysis. Solutions are illustrated for various initial conditions.

Low order practical Runge-Kutta-Nyström methods

Runge-Kutta-Nystrom formulas applicable to the general second order vector initial value problem,y″=f(x, y, y′), are presented. Two families of computational methods requiring five and six evaluations of the function,f, per integration step are derived. The methods consist of embedded formulas fo adjacent order for the solution and its first derivative. This permits a stepping strategy based on error estimates of all components of the numerical solution. From each family of methods a member considered to have good numerical properties has been Selected. Some comparisons of these specific new methods with conventional Runge-Kutta techniques have been made, and the Nystrom methods studied here seem to be competitive with some of the best Runge-Kutta methods currently in use.

Nonlinear waves on a coupled density front

Abstract It is shown that the inclusion of the nonlinear terms in the equations of motion of a coupled density front of zero potential vorticity results in wave solutions which merely propagate with time. The linear theory, on the other hand, predicts an exponential temporal growth. The nonlinear equation admits steady solutions representing standing waves whereas if the nonlinear terms are omitted no steady solutions exist. The general initial value problem is difficult to solve numerically since the linear problem is ill posed. In addition we prove that the general similarity solution of the nonlinear equation tends to zero for large times, at any point in space, regardless of the initial condition.