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1.2.2 Partial differential equations

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Partial differential equations (PDE) involve at least 2 variables in space (boundary value problems) or time (initial value problems). The equation is called second order if there are not more than two derivatives acting on the same unknown function $ f$

$\displaystyle A\frac{\partial^2 f}{\partial t^2} + 2B\frac{\partial^2 f}{\parti... ...l x^2} + D(t,x,\frac{\partial f}{\partial t},\frac{\partial f}{\partial x}) = 0$ (1.2.2#eq.1)

It is linear if $ D$ is linear in $ f$ , and is homogeneous if all the terms in $ D$ depend only on $ f$ . Initial value problems involving a combination of linear and non-linear operators

$\displaystyle \frac{\partial f}{\partial t}=\mathcal{L}f \approx \mathcal{L}_1 f+\mathcal{L}_2 f+\hdots +\mathcal{L}_m f$ (1.2.2#eq.2)

can often be solved with an operator splitting. The idea, based on the separation of variables, is to decompose the operator into a linear combination with $ m$ independent terms and to solve the problem in sub-steps where each part is evolved separately while keeping all the others fixed. Non-linearities are often split from linear terms that can be treated with the special techniques shown in this course (e.g. exercise 6.04). Multi-dimensional problems can sometimes be solved using a linear splitting called alternating-direction implicit (ADI), where only one spatial dimension is evolved at a time using an implicit scheme.

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