Date post: | 18-Nov-2015 |
Category: |
Documents |
Upload: | arun-govind-neelan-a |
View: | 6 times |
Download: | 0 times |
Numerical Methods for Partial Differential Equations CAAM 452Spring 2005Lecture 12
Instructor: Tim Warburton
Godunov Scheme SummaryTo complete this scheme we now specify how to compute the slopes.
Standard formulae:
With Limiting
Minmod slope limiter:
Monotonized central-difference limiter (MC limiter)
TodayMore limiters
Flux limiting function formulation.
We will discuss Hartens sufficient conditions for a numerical method (including limiter) to be TVD
Sweby TVD diagrams for flux limiting functions.
Extension to systems of linear PDEs
Extension to nonlinear PDEs
Flux Formulation with Piecewise Linear ReconstructionLast time we showed how the ansatz of a piecewise linear reconstruction and Godunovs method allowed us to compute the time averaged flux contribution at each end of the Ith cell
Notice: we can obtain the i-1/2 flux by setting i->i-1 in the i+1/2 flux formula(i.e. the flux formula is continuous at the cell boundary)
contUsing this notation the scheme becomes:
This is known as the flux formulation with piecewise reconstruction.
contBy writing the time interval averaged flux function in this way:
We are philosophically moving away from a local cell reconstruction approach towards controlling the flux contribution from jumps in the averages between elements.
Flux LimitersThe idea is: limit the flux of q between cells and you will subsequently limit spurious growth in the cell averages near discontinuities
A general approach is to multiply the jump in cell averages by a limiting function:
contThe theta ratio can be thought of as a smoothness indicator near the cell interface at x_{i-1/2}.
If the data is smooth we expect the ratio to be approximately 1 (except at extrema)
Near a discontinuity we expect the ratio to be far away from 1.
The flux limiting function, phi, will range between 0 and 2. The smaller it is, the more limiting is applied to a jump in cell averages. Above 1 it is being used to steepen the effective reconstruction.
contUsing this formulation we can recover the methods we have seen before and some new limiters:
contUsing this notation we can write down the scheme in terms of the flux limiter function ( ):
Upwind schemeflux contibutionLimited downwindcell interface fluxcontributionLimited upwind cell interfaceflux contributionu>0u
Hartens TheoremTheorem: Consider a general method of the form:
for one time step, where the coefficients C and D are arbitrary values (which in particular may depend on qbar in some way).
Then provided that the following conditions are satisfied:
Sweby Diagramshttp://locus.siam.org/fulltext/SINUM/volume-21/0721062.pdf
We need to express the flux limited scheme:
In the form:
And then satisfying the Harten conditions will guarantee the method is TVD.
An appropriate choice (which we can work with) is:
contIn this case since the D coefficients are zero and the Harten TVD conditions reduce to:
This will hold if:We can summarize this in terms of the minmod function:In addition we require:See LeVeque p 116-118 for details
cont
i.e. any flux limiting function must satisfy:
to be TVD. Graphically, the shaded region is the TVD region:
Clearly non of these linear limiters generate a TVD scheme.Lax-WendroffFrommBeam-Warming12312
contTo guarantee second order accuracy and avoid excessive compression of solutions, Sweby suggested the following reduced portion of the TVD region as a suitable range for the flux limiting function:
12312http://locus.siam.org/fulltext/SINUM/volume-21/0721062.pdf
Minmod Flux Limiter on Sweby Diagram
12312It is apparent that the minmod flux limiter applies the maximum possible limiting allowed within the second order TVD region.(i.e. it will be rather dissipative and smear out discontinuities somewhat as seen on the right hand side figure).
Superbee Flux Limiter on Sweby Diagram
12312The Superbee limiter applies the minimum limiting and maximum steepeningpossible to remain TVD. It is known to suffer from excessive sharpening ofslopes as a result. On the right we show what happens to a smooth sine wave after 20 periods.Notice the flattening of the peaks and the steepening of the slopes.
MC Flux Limiter on Sweby Diagram
12312The MC limiter transitions from upwind (theta
van Leer Flux Limiter
The van Leer limiter charts a careful compromise path throughthe Sweby TVD region.
Summary of Some Flux Limiting Functions
Nonlinear second orderTVD limitersLinear non-TVD limiters
ImplementationFor u>0 the scheme looks like:
We can easily achieve this in matlab:
Matlab Version
This is a sample Matlab implementation of a piecewise linear reconstructed Godunov approach with a selection of flux limiters.
Available from the course home page:
http://www.caam.rice.edu/~caam452/CodeSnippets/fluxlimiter.m
With the initial condition supplied by:
http://www.caam.rice.edu/~caam452/CodeSnippets/fluxlimiterexact.m
Homework 4Q1) Using N=80,160,320,640,1280 estimate the solution order of accuracy of theflux limited scheme:
with flux limiting functions: i. Fromm ii. minmod iii. MCusing initial conditions: i. sin(pi*x) ii. sin(pi*x) + (abs(x-.5)
Homework 4 contQ2a) Invent your own 2nd order TVD flux limiter function (i.e. a function with range contained in the Sweby TVD region)
Q2b) Modify sweby.m to plot your flux limiter function and compare with the limiter functions already used.
Q2c) Estimate order of accuracy for a smooth initial condition to the advection equation
Q2d) Estimate order of accuracy for a discontinuous initial condition to the advection equation
Q2e) Compare results (with diagrams,results and comments) and discuss how your limiter differs from the other limiters we have seen.