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Ames W.F. Numerical Methods for Partial Differential Equations

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Ames W.F. Numerical Methods for Partial Differential Equations
Second edition — Academic Press, 1977. — 376 p. — ISBN 978-1483235509
That part of numerical analysis which has been most changed by the ongoing revolution in numerical methods is probably the solution of partial differential equations. The equations from the technological world are often very complicated. Usually, they have variable coefficients, nonlinearities, irregular boundaries, and occur in coupled systems of differing types (say, parabolic and hyperbolic). The 'curse of dimensionality' is ever present- problems with two or three space variables, and time, are within our computational grasp.
Early development of calculational algorithms was based more upon the extension of methods for hand computation, empiricism, and intuition than on mathematical analyses. With increasing education and the subsequent development of the professional numerical analyst, the pattern is changing. New, useful methods are evolving which come closer to full utilization of the inherent powers of high-speed, large-memory computing machines. Many significant and powerful methods await discovery both for problems which are computable with existing techniques and those which are not. Unfortunately, as in other portions of mathematics, the abstract and the applications have tended to diverge. A new field of pure mathematics has been generated and while it has produced some results of value to users, the complexities of real problems have yet to be significantly covered by the presently available theorems. Nevertheless, guidelines are now available for the person wishing to obtain the numerical solution to a practical problem.
The present volume constitutes an attempt to introduce to upper-level engineering and science undergraduate and graduate students the concepts of modern numerical analyses as they apply to partial differential equations. The book, while sprinkled liberally with practical problems and their solutions, also strives to point out the pitfalls - e.g., overstability, consistency requirements, and the danger of extrapolation to nonlinear problems methods which have proven useful on linear problems. The mathematics is by no means ignored, but its development to a keen-edge is not the major goal of this work.
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