內(nèi)容40.流體力學(xué)numerical methods for heat,年-peakchina l

1、Lecture 25: Flux Limiters (Contd)Last TimeDeveloped concept of flux limitersIdentified the range of values these limiters can take Understood why Lax-Wendroff and Beam-Warming schemes do not satisfy upwind range conditionThis TimeDevelop specific flux limiter functionsExamine effect of these limiter

2、s on wave transportLook at multidimensional problems Flux Limiter Range(r) = 0 for r0 How to Choose (r)?Any general 5-point scheme involving WW,W, P, E, EE can be shown to be a linear combination of Lax-Wendroff and Beam-Warming SchemesThus, we choose limiter functions (r) which fall between the two

3、 bounding curves for these schemesFurthermore, we would like our (r) function to pass through the point (1,1).r=1 implies smooth variationFor this we would like to recover original second-order scheme How to Choose (r) (Contd) ?Another desirable property for symmetry reasons is:Summary: Any curve fa

4、lling between the bounding LW and BM schemes in the gray region which passes through (1,1) and has the symmetry property is a good curve to chooseLimiter Function Range for Second-Order SchemesMinmod Limiter FunctionThe minmod limiter is defined by:Superbee Limiter FunctionThe superbee limiter funct

5、ion is given by:Recall Explicit UDS SolutionStart with sine or square wave at t=0. Choose mesh of 50 cells and time step commensurate with =0.5. Take 25 time steps and plot result.Wave Transport: Minmod LimiterGenerally good, though slightly diffusiveWiggliness goneClipping of extrema evidentThis is

6、 not due to artificial diffusion, but due to clippingWave Transport: Superbee LimiterGenerally good performanceNo wigglesSlight steepening of curves evident in sine waveDiscussionThink of limiter as adding negative diffusion to the first-order upwind scheme Makes the scheme more accurate (second ord

7、er) but tends to promote wigglesMinmod limiter adds just enough negative diffusion to make scheme formally second orderStill somewhat diffusiveDiscussion (Contd)Superbee limiter adds more negative diffusion than minmodLess diffusive than minmodArtificial steepening is a precursor to formation of wig

8、glesSteady State ConsiderationsThough we have derived the limiter functions using unsteady wave transport arguments, it is possible to show that the same (r) functions are sufficient to guarantee non-wiggly solutions in steady stateThe face value e is found using:Using an upwind evaluation of the gr

9、adient, for example:Steady State Considerations (Contd)Convergence issues can e a problem when iterative schemes are usedDue to abrupt changes in slope in the (r) function for minmod and superbee limitersVariety of smoother functions have been proposed in the literatureVal Leer and Van Albada limite

10、rsQuadratic and cubic limitersVan Leer and Van Albada LimitersVan Leer limiter given by:Van Albada limiter given by:Van Leer and Van Albada LimitersQuadratic and Cubic LimitersThese are approximations to the minmod limiterQuadratic limiter:Cubic limiterSomewhat diffusive, like the minmod limiter Slope discontinuity at r=2Not as bad as minmod at r=1Quadratic and Cubic LimitersSince limiter falls somewhat outside the 2nd-order region, accuracy is somewhat less than second order for some values of rClosureIn this lecture, weDevelop