SYLLABUS Previous: 1.2.1 Ordinary differential equations
Up: 1.2 Differential Equations
Next: 1.2.3 Boundary / initial conditions
1.2.2 Partial differential equations
Slide : [ PDE
second order 
splitting 
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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

(1.2.2#eq.1) 
It is linear if
is linear in
, and is homogeneous if
all the terms in
depend only on
.
Initial value problems involving a combination of linear and nonlinear
operators

(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
independent terms and to
solve the problem in substeps where each part is evolved separately
while keeping all the others fixed.
Nonlinearities are often split from linear terms that can be treated
with the special techniques shown in this course (e.g. exercise 6.04).
Multidimensional problems can sometimes be solved using a linear
splitting called alternatingdirection implicit (ADI), where only one
spatial dimension is evolved at a time using an implicit scheme.
SYLLABUS Previous: 1.2.1 Ordinary differential equations
Up: 1.2 Differential Equations
Next: 1.2.3 Boundary / initial conditions
