e-book Numerical Solution of Partial Differential Equations: Finite Difference Methods

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The contents of this course is suitable for viewers at the graduate level, and is meant to serve as preparatory material for application-specific advanced computational courses such as computational fluid dynamics, computational heat transfer, and computational electromagnetics. The solution of PDEs can be very challenging, depending on the type of equation, the number of independent variables, the boundary, and initial conditions, and other factors. These two methods have been traditionally used to solve problems involving fluid flow.

The material is meant to serve as a prerequisite for students who might go on to take additional courses in computational mechanics, computational fluid dynamics, or computational electromagnetics. The notations, language, and technical jargon used can be easily understood by scientists and engineers who may not have had graduate-level applied mathematics or computer science courses.

Numerical Methods for Partial Differential Equations.

Numerical solution of a diffusion problem by exponentially fitted finite difference methods.

Barbara Zubik-Kowal. Eugene M. Finite Differences FD approximate derivatives by combining nearby function values using a set of weights. Several different algorithms are available for calculating such weights. Important applications beyond merely approximating derivatives of given functions include linear multistep methods LMM for solving ordinary differential equations ODEs and finite difference methods for solving partial differential equations PDEs.

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Figure 1. The open circle indicates a typically unknown derivative value, and the filled squares typically known function values. The most natural way to obtain FD weights is to require that the resulting formula becomes exact for as high degree polynomials as possible. On non-uniform grids of finite width, the derivative approximation at each node point requires a separate set of weights. If every stencil extends over all the node points, algorithms to calculate them can save operations by utilizing the fact that all the stencils are based on the same node set Weideman and Reddy The optimal FD weights can be calculated in just two lines of symbolic algebra code.

In Mathematica 7, these two lines are:. One way to introduce pseudospectral PS methods is to consider the limit of increasing orders. Total error : The best accuracy is usually obtained when these different error types match.

FD formulas for analytic functions : If a function is known to be analytic, and it can be computed also for complex arguments, Cauchy's integral formula leads to FD approximations that do not use values near the point along the real axis but instead on a circle in the complex plane around the point of approximation. In this case, high accuracy does not require h to be very small, and also high order derivatives say, 50th, or th become numerically available to full machine precision.

MIT Numerical Methods for PDE Lecture 3: Finite Difference for 2D Poisson's equation

With scattered nodes, one might try to generalize 5 to require the exact result for multivariate polynomials of increasing degrees. This approach fails because many perfectly reasonable node configurations will result in singular linear systems. A much more robust approach is to replace polynomials with radial basis functions RBFs , for example of Gaussian type.

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In 1-D and in a certain limit flat basis functions , this approach reproduces all previous 1-D results, but it generalizes reliably to scattered nodes in any number of dimensions. Applications of this emerging RBF-FD methodology not only include FD methods on irregularly shaped domains, but also on simpler ones such as the surface of a sphere. Without the need for a surface-bound coordinate system, complications such as pole singularities will not arise. With the approach just outlined often denoted RBF-FD in the literature , the stencil for approximating a derivative at a node will include a certain number of nearby nodes.

The kd-tree algorithm can very quickly identify sets of neighbors for each node from arbitrarily ordered or entirely unordered node lists. In particular, there is no need with RBF-FD implementations to create any unstructured grid that connects adjacent nodes to each other, nor to decompose the space into a collection of small separate elements.


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The last area is under rapid development, and will be described at a later time. The other two are summarized below.


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  • Other Scholarpedia articles provide surveys of a wide range of ODE ordinary differential equation solution techniques, both for initial value and for boundary value problems. In our present context, we limit our short ODE comments to LMM, since any one of these methods - of any order of accuracy - in principle only amounts to a quite immediate application of a single FD formula.

    All the main types of numerical methods for ODE initial value problems generalize immediately from a single scalar first order ODE to systems of any number of coupled such equation. Furthermore, ODEs of higher order can always be rewritten as such first order systems. Hence, we focus here only on solving.