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The reason for the faster convergence is that if we ignore the quadratic terms in (15 blood pressure medication images purchase generic metoprolol. Thus arteria tibial anterior order 50 mg metoprolol with amex, poor accuracy in time integration brings us to the root faster than solving the Newton flow accurately in pseudotime arteria maxilar discount metoprolol 50mg amex. When r(u) = 0 is the pseudospectral discretization of a differential equation heart attack toni braxton babyface proven metoprolol 25mg, the Jacobian matrix is dense and costly to invert. Fortunately, if the Newton flow is modified by replacing the inverse-Jacobian matrix by a matrix which is nonsingular at the roots of r(u), but is otherwise arbitrary, the roots of r(u) are still critical points of the differential equation. Not all approximations of the Jacobian matrix will succeed because the modifications may turn some roots from attractors into repellors. Suppose, however, the inverse-Jacobian is replaced by the inverse of a preconditioning matrix: 1 du = - r(u) dT H Linearization near a root gives -1 du = -H J(u - u0) + O (u - u0)2 dT ["Pre-conditioned Newton flow"] (15. If the error (u - u0) is decomposed into the eigenvectors of (1/H)J, then the error in different components must decay at rates ranging between exp(-T) and exp(-2. However, the Euler forward scheme does give very rapid convergence: with a time step of 4/7, one can guarantee that each component of the error will decrease by at least a factor of 2. Nevertheless, it is more than a little important to remember that there are no style points in engineering. Iterative methods, like mules, are useful, reliable, balky, and often infuriating. Note further that for constant coefficient ordinary differential equations and for separable partial differential equations, Galerkin/recurrence relation methods yield sparse matrices, which are cheap to solve using Guassian elimination. However, un-preconditioned iterations converge very slowly and are much less reliable because a single eigenvalue of the wrong sign will destroy them. Besides being the difference between convergence and divergence if the matrix is indefinite, elimination on the coarsest grid seems to improve the rate of convergence even for positive definite matrices. Good, reliable iterative methods for pseudospectral matrix problems are now available, and they have enormously extended the range of spectral algorithms. Iterations make it feasible to apply Chebyshev methods to difficult multi-dimensional boundary value problems; iterations make it possible to use semi-implicit time-stepping methods to computed wall-bounded flows. Nevertheless, this is still a frontier of intense research - in part because iterative methods are so very important to spectral algorithms - and the jury is still out on the relative merits of many competing strategies. This chapter is not the last word on iterations, but only a summary of the beginning. Chapter 16 the Many Uses of Coordinate Transformations "There are nine and sixty ways of constructing tribal lays" - R. In the next section, the Chebyshev polynomial-to-cosine change-of-variable greatly simplifies computer programs for solving differential equations. In this chapter, we concentrate on one-dimensional transformations, whose mechanics is the theme of Sec. When the flow has regions of very rapid change - near-singularities, internal boundary layers, and so on - maps that give high resolution where the gradients are large can tremendously improve efficiency. Finally, in the last part of the chapter, we give a very brief description of the new frontier of adaptive-grid pseudospectral methods. As simple and effective as these recurrences are, however, it is often easier to exploit the transformation x = cos(t) which converts the Chebyshev series into a Fourier cosine series: Tn (x) cos(nt) (16. Similarly 1 d2 Tn (x) sin(t) -n2 cos(nt) - cos(t) [-n sin(nt)] = dx2 sin3 (t) (16. However, this is irrelevant to the Chebyshev "roots" grid because all grid points are on the interior of the interval. The problem also disappears for the "extrema" (GaussLobatto) grid if the boundary conditions are Dirichlet, that is, do not involve derivatives. For example, uxx - q u = f (x) becomes sin(t) utt - cos(t) ut - q sin3 (t) u = sin3 (t) f (cos[t]) u(0) = u = 0 t [0,] (16. My personal preference is to solve the problem in x, burying the trigonometric formulas in the subroutines that evaluate derivatives.
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For the KdV equation xeloda arrhythmia order metoprolol 12.5mg amex, the separation is spatial: the dispersing transients travel leftward (because the group velocity for small amplitude waves of all wavenumbers is negative) whereas the velocities of all solitary waves are positive heart attack diagnosis purchase metoprolol overnight delivery. Because the amplitude of a solitary wave decreases exponentially fast with distance from the center of the soliton arteria spinalis anterior order metoprolol overnight, the overlap between a given solitary wave and the rest of the solution decreases exponentially with time blood pressure chart age 13 metoprolol 12.5mg low cost. Thus, even though there is no instability and the eigenvalue problem is nonlinear, the power method converges geometrically for the KdV equation. The inverse power method, also sometimes called simply "inverse iteration", lacks this limitation. It follows that the inverse power method will converge geometrically to any eigenvalue of the matrix A, provided we have a sufficiently good first guess (and set equal to it). However, as explained in Chapter 15, one can often solve the matrix equation implicitly (and cheaply) through a preconditioned Richardson iteration instead. The inverse power method is very powerful because it can be applied to compute any eigenmode. Indeed, the routines in many software libraries compute the eigenfunctions by the inverse power method after the eigenvalues have been accurately computed by a different algorithm. The power and inverse power algorithms are but two members of a wide family of iterative eigenvalue solvers. The Arnoldi method, which does not require storing the full matrix A, is particularly useful for very large problems. The so-called "neutral curve", which is the boundary between stability and instability in parameter space, is usually an important goal. In the next section, we describe a good strategy for computing the eigenvalues even when the differential equation is singular. Many an arithmurgist has traced one branch of unstable modes with loving care, only to find, years later, that another undiscovered mode had faster growth rates in at least part of the parameter space. These singularities, usually called "critical latitudes" or "critical points", create severe numerical difficulties. However, a good remedy (Boyd, 1985a) is to make a transformation from the original variable y such that the problem is solved on a curve in the complex y-plane rather than on the original interval on the real yaxis. With the proper choice of map parameter, one can loop the curve of integration away from the singularity so that it does not degrade numerical accuracy. If a and b were of the same sign so that the singularity was not on the interior of the interval, y [a, b], then (7. As it is, not only the differential equation but also the solution are singular on the interior of the interval. After we have made a simple linear stretching to shift the interval from [a, b] to [-1, 1], an effective transformation for (7. Because of the change of variable, however, the real interval in x is an arc in the complex y-plane which detours away from the singularity. Since u(y) has a branch point at y = 0, the choice of looping the contour above or below the real y-axis is an implicit choice of branch cut. The correct choice can be made only by a careful physical analysis; for geophysical problems, Boyd (1982c) explains that the proper choice is to go below the real axis by choosing > 0 in (7. Boyd (1981b) solved this same problem using an artificial viscosity combined with an iterative finite difference method - and missed two eigenvalues with very small imaginary parts (Modes 3 and 7 in the table). It is also a method with a weakness in that the Chebyshev series for u(y) converges most rapidly for real x - but this is an arc of complex y. The series converges more and more slowly as we move away from the arc that corresponds to real x and it must diverge at the singularity at y = 0. Therefore, the detour into the complex plane is directly useful only for computing the eigenvalues. Once we have, of course, we can use a variety of methods to compute the corresponding eigenfunction. The errors for N < 40 are the differences from the results for N = 40 for the modes shown. Modes that are wildly in error or violate the theorem that the imaginary part of the eigenvalue is always positive have been omitted; thus only one eigenvalue is listed for N = 6 although the Chebyshev-discretized matrix eigenproblem had five other eigenvalues.
Eigenvalues 11 to 16 are way off hypertension before pregnancy metoprolol 12.5mg with amex, and there is no point in accurately tracking the time evolution of these modes of the Chebyshev pseudospectral matrix because they have no counterpart in the solution of the diffusion equation arrhythmia causes purchase metoprolol. The slow manifold is composed of the modes left of the vertical dividing line whose Chebyshev eigenvalues do approximate modes of the diffusion equation blood pressure vs blood sugar purchase metoprolol online pills. The largest numerical eigenvalue is about 56 blood pressure of 150/90 buy metoprolol 100 mg line,000 whereas the largest accurately approximated eigenvalue is only about 890, a ratio of roughly 64, and one that increases quadratically with N. The exact solution to the diffusion equation is of the same form except that the eigenvectors are trigonometric functions and the eigenvalues are different as given by the first line of Table 12. The good news of the table is: the first (2/)N eigenvalues are wellapproximated by any of the three spectral methods listed. This implies that we can follow the time evolution of as many exact eigenmodes as we wish by choosing N sufficiently large. The bad news is: the bad eigenvalues are really bad because they grow as O(N 4) even though the true eigenvalues grow only as O(N 2). The stability limit (set by the largest eigenvalue) is very small compared to the time scale of the mode of smallest eigenvalue. The slow manifold is the span of the eigenmodes which are accurately approximated by the discretization, which turns out to be the first (2/)N modes for this problem. However, the physics is also relevant: if only the steady-state solution matters, than the slow manifold is the steady-state, and the time evolution of all modes can be distorted by a long time step without error. The physics and numerics must be considered jointly in identifying the slow manifold. The solid curve is not the result of a running time average, but rather is the forecast from the initial conditions obtained by adjusting the raw data onto the slow manifold. Unfortunately, raw observational data is corrupted by random instrumentation and sampling errors which, precisely because they are random, do not respect quasi-geostrophic balance. Initializing a forecast model with raw data generates a forecast with unrealistically large gravity wave oscillations superimposed on the slowly-evolving large-scale dynamics (Daley, 1991, Lynch, 1992. The remedy is to massage the initial conditions by using an algorithm in the following category. Meteorological simulations of large-scale weather systems can be subdivided into forecasting and climate modelling. For the latter, the model is run for many months of time until it "spins up" to statistical equilibrium. The choice of initial conditions is irrelevant because the flow will forget them after many months of integration; indeed, independence of the initial conditions is an essential part of the very definition of climate. The inherent dissipation of the model gradually purges the fast transients, like white blood cells eating invading germs. Similarly, the backwards Euler scheme, which damps all frequencies but especially high frequencies, will pull the flow onto the slow manifold for almost any physical system, given long enough to work. When the flow must be slow from the very beginning, however, stronger measures are needed. Modifying the partial differential equations so that the approximate equations support only "slow" motions (quasi-geostrophic and various "balanced" systems, reviewed in Daley, 1991). Method of multiple scales ["Normal mode initialization"] (Machenhauer, 1977, Baer, 1977, Baer and Tribbia, 1977). Without assessing the relative merits of these schemes for operational forecasting, we shall describe the method of multiple scales, or "normal mode initialization" as meteorologists call it, because it provides the theoretical underpinning for the Nonlinear Galerkin methods. F = = fS (S, F) fF (S, F) after partitioning the solution u into its slow and fast modes, S and F, respectively. S is a vector whose j-th element is the product of the frequency of the j-th mode. The obvious way to integrate on the slow manifold is to simply ignore the fast modes entirely and solve the reduced system St + i S. The nonlinear interaction of the slow modes amongst themselves will create a forcing for the fast modes.
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