From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: adaptive upwinding
Date: 8 Mar 2000 12:07:28 GMT
Newsgroups: sci.math.num-analysis
Summary: Courant Isaacson Rees algorithm for solving differential equations

In article <38C523BE.1BAE0442@gre.ac.uk>,
 Sweta Rout <s.rout@gre.ac.uk> writes:
|> can somebody give me good references for the adaptive upwinding
|> algorithm?  if u know it off the top of ur head, could u write it down
|> too?  thanks.
|> 
maybe you mean courant_isaacson_rees?

differential equation:
 (d/dt) u(x,t) = A(x) (d/dx) u(x,t)  , u(x,o)=phi(x) ,
  x in [x_u,x_o] , t>=0 
  with A(x) real diagonalizable with n different eigenvalues 
(which can be solved in the dependency-region of the initial values only)
choose  
 delta_t such that l = (definition ) = delta_t/delta_x 
       satisfies  l*sup_x ||A(x)|| <=1
   
  u_{i,j+1} = l A(-)(x_i)u_{i-1,j} +
              ( I -l(A(-)(x_i)+A(+)(x_i) )u_{i,j} + lA(+)(x_i)u_{i+1,j} 

here M(x) eigenvector matrix of A(x), D(x) eigenvalue-matrix with eigenvalues 
lambda_i(x),
D(+)_{ii} = lambda_i(x) if lambda_i(x)>0, =0 otherwise
D(-)_{ii} = lambda_i(x) if lambda_i(x)<0, =0 otherwise
A(+)(x) = M(x) D(+)(x) M^{-1}(x)  
A(-)(x) = M(x) D(-)(x) M^{-1}(x)

hope that helps
peter
==============================================================================                
From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: adaptive upwinding
Date: 9 Mar 2000 09:59:05 GMT
Newsgroups: sci.math.num-analysis

In article <8a5lo2INNplb@subds.rzg.mpg.de>,
 bds@rzg.mpg.de (Bruce Scott TOK) writes:
|> In article <8a5fq0$oad$1@sun27.hrz.tu-darmstadt.de>,
|> Peter Spellucci <spellucci@mathematik.tu-darmstadt.de> wrote:
|> 
|> >maybe you mean courant_isaacson_rees?
|> 
|> [very nice description]
|> 
|> Can you give the standard reference?
I took it from 
Meis&Marcowitz: Numerische Behandlung partieller Differentialgleichungen, 
Springer 1978 (in German) but it is also in 
Richtmyer&Morton: Difference methods for initial value problems,
page 290.
The idea is "let the difference star follow the direction(s) of
the characteristics", best visualized for the case of a scalar (i.e. a convection)
equation
peter