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Analysis of numerical dissipation and dispersion - Fakultät

Analysis of numerical dissipation and dispersionModified equation method:the exact solution of the discretized equationssatisfies a PDE which is generally different from the one to be solvedOriginal PDEM odified equationAun+1=Bun u t+Lu= 0 u t+Lu= Xp=1 2p 2pu x2p+ Xp=1 2p+1 2p+1u x2p+1 Motivation: PDEs are difficult or impossible to solve analytically but theirqualitative behavioris easier to predict than that of discretized equations Expand all nodal values in the difference scheme in a double Taylor seriesabout a single point (xi, tn) of the space-time mesh to obtain a PDE Express high-order time derivatives as well as mixed derivatives in termsof space derivatives usingthisPDE to transform it into the desired formDerivation of the modified convection equation u t+v u x= 0, v >0 BDS in space, FE in time:un+1i uni t+vuni uni 1 x= 0 (upwind)Taylor series expansions about the point (xi, tn)un+1i=uni+ t u t ni+( t)22 2u t2 ni+( t)36 3u t3 ni+.

Analysis of numerical dissipation and dispersion Modified equation method: the exact solution of the discretized equations satisfies a PDE which is generally different from the one to be solved

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Transcription of Analysis of numerical dissipation and dispersion - Fakultät

1 Analysis of numerical dissipation and dispersionModified equation method:the exact solution of the discretized equationssatisfies a PDE which is generally different from the one to be solvedOriginal PDEM odified equationAun+1=Bun u t+Lu= 0 u t+Lu= Xp=1 2p 2pu x2p+ Xp=1 2p+1 2p+1u x2p+1 Motivation: PDEs are difficult or impossible to solve analytically but theirqualitative behavioris easier to predict than that of discretized equations Expand all nodal values in the difference scheme in a double Taylor seriesabout a single point (xi, tn) of the space-time mesh to obtain a PDE Express high-order time derivatives as well as mixed derivatives in termsof space derivatives usingthisPDE to transform it into the desired formDerivation of the modified convection equation u t+v u x= 0, v >0 BDS in space, FE in time:un+1i uni t+vuni uni 1 x= 0 (upwind)Taylor series expansions about the point (xi, tn)un+1i=uni+ t u t ni+( t)22 2u t2 ni+( t)36 3u t3 ni+.

2 Uni 1=uni x u x ni+( x)22 2u x2 ni ( x)36 3u x3 ni+..Substitution into the difference scheme yields u t ni+v u x ni= t2 2u t2 ni ( t)26 3u t3 ni+v x2 2u x2 ni v( x)26 3u x3 ni+..original PDEO[ t, x]truncation error( )Next step: replace both time derivatives in the RHS by space derivativesDerivation of the modified equationDifferentiate ( ) with respect tot 2u t2+v 2u x t= t2 3u t3 ( t)26 4u t4+v x2 3u x2 t v( x)26 4u x3 t+..(1)Differentiate ( ) with respect toxand multiply byvv 2u t x+v2 2u x2= v t2 3u t2 x v( t)26 4u t3 x+v2 x2 3u x3 v2( x)26 4u x4+..(2)Subtract (2) from (1) and drop high-order terms 2u t2=v2 2u x2+ t2h 3u t3+v 3u t2 x+O( t)i+ x2hv 3u x2 t v2 3u x3+O( x)i(3)Differentiate formula (3) with respect tot 3u t3=v2 3u x2 t+O[ t, x](4)Differentiate formula (2) with respect tox 3u x2 t= v 3u x3+O[ t, x](5)Differentiate formula (3) with respect tox 3u t2 x=v2 3u x3+O[ t, x](6)Derivation of the modified equationEquations (4) and (5) imply that 3u t3= v3 3u x3+O[ t, x](7)Plug (5) (7) into (3) 2u t2=v2 2u x2+v2(v t x) 3u x3+O[ t, x](8)Substitute (7) and (8) into ( ) to obtain the modified equation u t+v u x= v2 t2h 2u x2+ (v t x) 3u x3i+v3( t)26 3u x3+v x2 2u x2 v( x)26 3u x3+.

3 Which can be rewritten in terms of the Courant number =v t xas follows u t+v u x=v x2(1 ) 2u x2|{z} numerical diffusion+v( x)26(3 2 2 1) 3u x3|{z} numerical dispersion +.. CFL stability condition 1 must be satisfied for the discreteproblem to be well-posed. In the case >1, the numerical diffusion coefficientv x2(1 ) is negative, which corresponds to abackward heat equationSignificance of terms in the modified equationExact solution of the discretized equationsAun+1=Bun u t+Lu= Xp=1 2p 2pu x2p+ Xp=1 2p+1 2p+1u x2p+1 Even-order derivatives 2pu x2pcause numerical (amplitude errors)Odd-order derivatives 2p+1u x2p+1cause numerical (phase errors) u t+v u x= 0 Qualitative Analysis : the numerical behavior of the discretization scheme largelydepends on the relative importance of dispersive and dissipative effectsStabilization by means of artificial diffusionStability condition (necessary but not sufficient)The coefficients of the even-order derivatives in the modifiedequation musthave alternating signs, the one for the second-order term being positiveIf this condition is violated, it can be enforced by adding artificial diffusion.

4 Stabilized methods+ (v )2ustreamline diffusionNonoscillatory methods+ (v )2u+ (u) ushock-capturing the one-dimensional case both terms are proportional to 2u x2 Free parameters =c h1+|v|, (u) =c h2R(u)wherehis the mesh sizeandR(u) is the residualProblem: how to determine proper values of the constantsc andc ???Alternative: use a high-order time-stepping method or flux/slope limitersLax-Wendroff time-steppingConsider a time-dependent PDE u t+Lu= 0 in (0, T)1. Discretize it in time by means of the Taylor series expansionun+1=un+ t u t n+( t)22 2u t2 n+O( t)32. Transform time derivatives into space derivatives using the PDE u t= Lu, 2u t2= t u t = t( Lu) = L u t=L2u3. Substitute the resulting expressions into the Taylor seriesun+1=un tLun+( t)22L2un+O( t)34. Perform space discretization using finite differences/volumes/elementsLax-Wendroff scheme for pure convection equation u t+v u x= 0 (1D case)Time derivativesL=v x u t= v u x, 2u t2=v2 2u x2 Semi-discrete schemeun+1=un v t u x n+(v t)22 2u x2 n+O( t)3 Central difference approximation in space u x i=ui+1 ui 12 x+O( x)2, 2u x2 i=ui+1 2ui+ui 1( x)2+O( x)2 Fully discrete scheme (second order in space and time)un+1i uni t+vuni+1 uni 12 x=v2 t2uni+1 2uni+uni 1( x)2+O[( t)2,( x)2] is equivalent to FE/CDS stabilized by numerical dissipationdue to the second-order term in the Taylor series (no adjustable parameter)Forward Euler vs.

5 Lax-Wendroff (CDS)Modified equation for the FE/CDS scheme u t+v u x= v x2 2u x2 v( x)26(1 + 2 2) 3u x3+..where =v t x unconditionally unstable since the coefficient v x2 = v2 t2is negativeModified equation for the LW/CDS scheme u t+v u x= v( x)26(1 2) 3u x3 v( x)38 (1 2) 4u x4 v( x)4120(1+5 2 6 4) 5u x5+.. conditionally stable for 2 1 in 1D, 2 18in 2D, 2 127in 3D the second-order derivative (leading dissipation error) has been eliminated the negative dispersion coefficient corresponds to a lagging phase error i. e. harmonics travel too slow, spurious oscillations occurbehindsteep fronts the leading truncation error vanishes for 2= 1 (unit CFL property)Forward Euler vs. Lax-Wendroff (FEM)Galerkin FEM u t i Mun+1i uni t,whereMui=ui+1+4ui+ui 16 Modified equation for the FE/FEM scheme u t+v u x= v x2 2u x2 v( x)23 2 3u x3+..where =v t x unconditionally unstable since the numerical diffusion coefficient is negative the leading dispersion error due to space discretization has been eliminatedModified equation for the LW/FEM scheme u t+v u x=v( x)26 2 3u x3 v( x)324 (1 3 2) 4u x4+v( x)4180(1 152 2+ 9 4) 5u x5+.

6 Conditionally stable for 2 13in 1D, 2 124in 2D, 2 181in 3D the positive dispersion coefficient corresponds to a leading phase error i. e. harmonics travel too fast, spurious oscillations occuraheadof steep fronts the truncation error does not vanish for 2= 1 (no unit CFL property)Lax-Wendroff FEM in multidimensionsPure convection equation u t+v u= 0 in (0, T)v=v(x)Boundary conditionsu=gon in={x :v n<0}inflow boundaryTime derivativesL=v u t= v ustreamline derivative 2u t2= (v )2ustreamline diffusion (second derivative in the flow direction)Semi-discrete schemeun+1=un tv un+( t)22(v )2un+O( t)3 Weak formulation for the Galerkin methodZ w(un+1 un)dx= tZ wv undx+( t)22Z w(v )2undxIntegration by parts using the identity (ab) =a b+b ayieldsR wv v udx= R (wv)v udx+R outwv nv uds= R v wv u dx R w v v u dx+R outwv n v u dsTaylor-Galerkin methodsDonea (1984)introduced a family of high-order time-stepping schemes whichstabilize the convective terms by means of intrinsic streamline diffusionConvection-dominated PDE u t+Lu= 0 in (0, T)Taylor series expansion up to the third orderun+1=un+ t u t n+( t)22 2u t2 n+( t)36 3u t3 n+O( t)4 Time derivatives u t= Lu, 2u t2= t u t = t( Lu) = L u t=L2u 3u t3=L2 u t=L2un+1 un t+O( t)

7 To avoid third-order space derivativesSubstitutionun+1=un tLun+( t)22L2un+( t)26L2(un+1 un)+O( t) Lax-Wendroff scheme is recovered forun+1=un(steady state)Euler Taylor-Galerkin schemeSemi-discrete FE/TG schemehI ( t)26L2iun+1 un t= Lun+ t2L2unSpace discretization: Galerkin FEM (finite differences/volumes also feasible)The third-order term results in a modification of the consistent mass convection in 1D u t+v u x= 0,L=v xModified equation for the FE/TG scheme (Galerkin FEM, linear elements) u t+v u x= v( x)324 (1 2) 4u x4+v( x)4180(1 5 2+ 4 4) 5u x5+.. conditionally stable for 2 1 in 1D, 2 18in 2D, 2 127in 3D the leading dispersion error is of higher order than that forLW/FEM the leading truncation error vanishes for 2= 1 (unit CFL property)Leapfrog Taylor-Galerkin schemeTaylor seriesun 1=un t u t n+( t)22 2u t2 n ( t)36 3u t3 n+O( t)4It follows thatun+1 un 1= 2 t u t n+( t)33 3u t3 n+O( t)4 Time derivatives u t= Lu, 3u t3=L2 u t=L2un+1 un t+O( t)Semi-discrete LF/TG schemehI ( t)26L2iun+1 un 12 t= LunModified equations for leapfrog schemes withL=v xLF/CDS u t+v u x= v( x)26(1 2) 3u x3+v( x)4120(1 10 2+ 9 4) 5u x5+.

8 LF/FEM u t+v u x=v( x)26 2 3u x3+v( x)4360(2 27 4) 5u x5+..LF/TG u t+v u x=v( x)4360(2 + 5 2 7 4) 5u x5+.. fourth-order accurate, non-dissipative and conditionally stable for 2 1 the truncation error shrinks as compared to that for 2nd-order LF schemes the unit CFL property is satisfied for phase angles in the range 0 2 Crank-Nicolson Taylor-Galerkin schemeTaylor series expansions up to the fourth orderun+1=un+ t u t n+( t)22 2u t2 n+( t)36 3u t3 n+O( t)4un=un+1 t u t n+1+( t)22 2u t2 n+1 ( t)36 3u t3 n+1+O( t)4It follows thatun+1=un+ t2h u t n+ u t n+1i+( t)24 2u t2 n 2u t2 n+1 +( t)312 3u t3 n+ 3u t3 n+1 +O( t)4 Time derivatives u t= Lu, 2u t2= t u t = t( Lu) = L u t=L2u 3u t3 n+ 3u t3 n+1=L2h u t n+ u t n+1i= 2L2un+1 un t+O( t)Fourth-order accurate Crank-Nicolson time-steppingun+1=un t2L(un+un+1) +( t)24L2(un un+1) +( t)26L2(un+1 un)Crank-Nicolson Taylor-Galerkin schemeSemi-discrete CN/TG schemehI+ t2L+( t)

9 212L2iun+1 un t= LunModified equations for Crank-Nicolson schemes withL=v xCN/CDS u t+v u x= v( x)26 1 + 22 3u x3+v( x)4120(1 + 5 2+32 4) 5u x5+..CN/FEM u t+v u x= v( x)212 2 3u x3+..CN/TG u t+v u x=v( x)4720(4 5 2+ 4) 5u x5+.. fourth-order accurate, non-dissipative and unconditionally stable cannot be operated at 2>1 since the matrix becomes singular the phase response is far superior to that for 2nd-order CN schemes the leading truncation error vanishes for 2= 1 (unit CFL property) LF/TG and CN/TG degenerate into the unstable Galerkindiscretization if the solution reaches a steady state so thatun+1=unMultistep Taylor-Galerkin schemesFractional step algorithmsof predictor-corrector type lend themselves to thetreatment of (nonlinear) problems described by PDEs of complexstructurePurpose: to avoid a repeated application of spatial differential operators tothe governing equation and/or enhance the accuracy of time discretizationTaylor seriesun+1=un+ t u t n+( t)22 2u t2 n+O( t)3 FactorizationI+ t t+( t)22 2 t2=I+ t t I+ t2 t Richtmyer scheme (two-step Lax-Wendroff method)un+1/2=un+ t2 u t nun+1=un+ t u t n+1/2 un+1/2=un t2 Lunun+1=un tLun+1/2 second-order RK method (forward Euler predictor + midpoint rule corrector) stability and phase characteristics as for the single-stepLax-Wendroff schemeMultistep Taylor-Galerkin schemesTaylor seriesun+1=un+ t u t n+( t)22 2u t2 n+( t)36 3u t3 n+O( t)4 FactorizationI+ t t+( t)22 2 t2+( t)36 3 t3=I+ t thI+ t2 t+( t)26 2 t2i=I+ t t I+ t2 t I+ t3 t no high-order derivativesThree-step Taylor-Galerkin method(Jiang and Kawahara, 1993)

10 Un+1/3=un+ t3 u t nun+1/2=un+ t2 u t n+1/3un+1=un+ t u t n+1/2 un+1/3=un t3 Lunun+1/2=un t2 Lun+1/3un+1=un tLun+1/2 third-order time-stepping method, conditionally stable for 2 1 (optimal) no improvement in phase accuracy as compared to the two-step TG algorithm lagging phase error at intermediate and short wavelengths, unit CFL propertyHigh-order Taylor-Galerkin schemesMultistep TG methods involving second time derivatives offer high accuracyand an isotropic stability domain for nonlinear multidimensional problemsTwo-step third-order TG scheme(Selmin, 1987)un+1/2=un+ t3 u t n+ ( t)2 2u t2 npredictorun+1=un+ t u t n+( t)22 2u t2 n+1/2corrector is chosen so as to obtain the desired stability/accuracy characteristics excellent phase response of the FE/TG method is reproduced for =19 stable for 2 34in 1D/2D/3D (no loss of stability in multidimensions)Underlying factorization vs. Taylor series expansionI+ t t+( t)22 2 t2hI+ t3 t+ ( t)2 2 t2i=I+ t t+( t)22 2 t2+( t)36 3 t3+ ( t)42 4 fourth-order accurate time-stepping method is recovered for =112 Two-step fourth-order TG schemesTTG-4A scheme(Selmin and Quartapelle, 1993)un+1/2=un t3 Lun+( t)212L2unpredictorun+1=un tLun+( t)22L2un+1/2corrector fourth-order accurate in time, isotropic stability condition 2 1 poor phase response at intermediate and short wavelengths as| | 1 TTG-4B scheme , , +1/2=un tLun+ ( t)2L2unpredictorun+1=un tLun+1/2+ ( t)2L2un+1/2corrector fourth-order accurate in time, isotropic stability condition 2 excellent phase response in the whole range of Courant numbersSemi-implicit Taylor-Galerkin schemesProblem: fully explicit schemes are doomed to be conditionallystableSemi-implicit Lax-Wendroff method(Hassan et al.)


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