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A characteristic curve of a PDE is a curve along which the initial value
problem of this PDE cannot be solved uniquely or not at all (locally).
The method of characteristics is based on the fact that a hyperbolic
PDE can be transformed into a system of ODE's determining simultaneously
its characteristic curves and the solution of the PDE along these curves.
Initial information of the solution is moved along the characteristic curves.
Here we use Eulers method to solve this system of ODE's.
We present only one fixed example with analytically known solution here.
- The PDE reads u=(u1,u2)
(d/dy)u1 = (d/dx)u2
(d/dy)u2 = f1(u)(d/dx)u1 + f2(u)(d/dx)u2 + f3(u)
with initial values u(x,0) = ( 0 , exp(2x) ).
f1(u)=-(1-u12)/(1-u22)
f2(u)=2u1u2/(1-u22)
f3(u)=-4u1exp(2x)/(1-u22)
and the exact solution
u(x,y)=2exp(x)(sin(y) , cos (y) ).
- We restrict x to [0,1]. The interval [0,1]
has as domain of determination a curvilinear triangle in the (x,y) plane
over which the solution will be computed (the solution components depend
exclusively on the initial values over [0,1]) .
- This PDE is quasilinear. Hence the characteristic curves depend on the solution u(x,y).
- We begin with an equidistant grid on [0,1]×{0}. Through each point (x,0)
there go two characteristic curves and we compute the crossings of these curves for a small
increment in y direction and the solution u there. These crossings lie
on a new curve, which is our next initial curve and we repeat this process. Clearly we
are loosing two outer grid points with every such step and finally end with one point
approximating the top of the (curvilinear) triangle which is the domain of determination of
our initial interval [0,1].
- The advantage of the method of characteristics is the fact that it
can approximate the analytical domain of determination quite good, and this
easily. The satisfaction of the CFL is inherent in its construction and not
an additional requirement as for the finite difference methods.
- Its disadvantage, that for satisfactory precision the computational effort
can be quite large and that in more than two dimensions (where characteristic surfaces
replace the curves) its application is practically impossible.
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