Transcription of Lecture Notes 4 Convergence Theory for Linear Methods
{{id}} {{{paragraph}}}
Example: biology
Lecture Notes 4 Convergence Theory for Linear Methods
Lecture Notes 4 Convergence Theory for Linear MethodsLetqnjbe the numerical approximation of the exact cell average,qnj unj:=1 x xj+1xju(tn, x)dx,tn=n t.(1)We want to check Convergenceqnj unjas x, t 0, Accuracy and Convergence rateqnj=unj+O( xp+ tr),for somep, r consider two cases of the numerical approximation: with and without boundaries. Whenthere are boundaries, we letqnbe the finite length vectorqn= (qn0, .. , qnN) there are no boundaries we letqndenote the infinite sequenceqn= (.. , qn 1, qn0, qn1..),and similarly for the exact solutionun. We write the numerical scheme compactly as an operatorNacting onqn,qn+1=N(qn, t, x).
x 0 (x,t) x−at=x 0 t (a) Continuous problem x j x j−n x j+n (x j,t n) n (b) Numerical approximation Figure 1. The CFL condition. 2 Checking stability Checking stability of a scheme is usually the most difficult part when proving convergence.
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document:
Related search queries
REAL ANALYSIS LECTURE NOTES, Convergence, Lecture notes, NONLINEAR PROGRAMMING LECTURE 4, LECTURE 4 CONVERGENCE, Lecture, Convergence and Divergence, Convergence and Divergence Lecture Notes, Economic Growth, Convergence of a Sequence, Monotone sequences, Lecture 18 : Improper integrals, Lecture 4 | September 11 4.1 Gradient Descent, Lecture 4 | September 11