Numerical Partial Differential Equations for Environmental Scientists and Engineers

This book concerns the practical solution of Partial Differential Equations. We assume the reader knows what a PDE is - that he or she has derived some, and solved them with the limited but powerful arsenal of analytic techniques. We also assume that (s)he has gained some intuitive knowledge of their solution properties, either in the context of specific applications, or in the more abstract context of applied mathematics. We assume the reader now wants to solve PDE's for real, in the context of practical problems with all of their warts - awkward geometry, driven by real data, variable coefficients, nonlinearities - as they arise in real situations. The applications we envision span classical mathematical physics and the "engineering sciences" : fluid mechanics, solid mechanics, electricity and magnetism, heat and mass transfer, wave propagation. Of course, these all share a joyous interdisciplinary unity in PDE's. The material arises from lectures at Dartmouth College for first-year graduate students in science and engineering. That audience has shared the above motivations, and a mathematical background including: ordinary and partial differential equations; a first course in numerical an- ysis; linear algebra; complex numbers at least at the level of Fourier analysis; and an ability to program modern computers. Some working exposure to applications of PDE's in their research or practice has also been a common denominator. This classical undergraduate preparation sets the stage for our "First Practical Course". Naturally, the "practical" aspect of the course involves computation.

The Finite Difference Method. - Finite Difference Calculus. - Elliptic Equations. - Iterative Methods for Elliptic Equations. - Parabolic Equations. - Hyperbolic Equations. - The Finite Element Method. - General Principles. - A 1-D Tutorial. - Multi-Dimensional Elements. - Time-Dependent Problems. - Vector Problems. - Numerical Analysis. - Inverse Methods. - Inverse Noise, SVD, and Linear Least Squares. - Fitting Models to Data. - Dynamic Inversion. - Time Conventions for Real-Time Assimilation. - Skill Assessment for Data Assimilative Models. - Statistical Interpolation.