Mixed and Implicit Schemes Implicit Schemes. Exercise: Verify that ρ is unimodular: ρ = 1.

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1 Mixed ad Implicit Schemes 3..4 The leapfrog scheme is stable for the oscillatio equatio ad ustable for the frictio equatio. The Euler forward scheme is stable for the frictio equatio but ustable for the oscillatio equatio. Suppose we require a approximatio to the equatio du = iωu κu, dt This is a prototype of the N.S equatios, with terms of both types. Oe approach is to use the leapfrog scheme for the oscillatio term ad the forward scheme for the frictio term: U + = U + iωu κu ). We ca show that this is stable provided κ + ω ). The strogest costrait imposed by the CFL criterio is for the highest wave-speed c occurrig i the system. For the atmosphere, the speed of exteral gravity waves may be estimated as c = ū + gh where ū is the advectig speed ad H is the scale height. We may assume ū < 00 m s ad gh < 300 m s, so a safe maximum value for c is 400 m s. The the maximum allowable time step for various spatial grid sizes may be estimated: : 00 km 00 km 0 km 0 km : 500 s 50 s 50 s 5 s I the two-dimesioal case, the stability criterio is more striget: we eed to choose a time step that is times smaller tha that permitted i the oe-dimesioal case. Moder umerical models of the atmosphere typically combie several distict schemes i this way. The Navier-Stokes equatios are V t + V V + Ω V + ρ p = ν V + g. They have advectio terms correspodig to the oscillatio equatio ad diffusio terms like the frictio equatio. We will shortly cosider the semi-implicit scheme, where some terms are itegrated implicitly ad others explicitly. NWP models also use various filterig processes to limit spatial ad temporal oise. Some of these represet diffusive physical processes. Others are just umerical dampig, to prevet spurious oise. Implicit Schemes For the simple oscillatio equatio du dt = iωu the cetered) implicit approximatio is ) U + U U + + U = iω. This is secod-order accurate ad ucoditioally stable: ) U + = ρ U + where ρ = iω iω I this simple case, we may solve immediately for U +. I geeral, we must solve a complicated oliear system. Exercise: Verify that ρ is uimodular: ρ =.

2 The Semi-implicit Method itro.) It is commo practice today to treat selected liear terms implicitly ad the remaiig terms explicitly. The semi-implicit method was pioeered by Adré Robert. The terms that give rise to high frequecy gravity waves are itegrated implicitly, eablig the use of a log time step. Formally, we separate the terms ito two groups. Thus, the equatio du dt = F u) = F u) + F u) is discretised by somethig like ) U + U = F U U + U + ) + F Schemes of this sort are pivotal i moder NWP models, due to their excellet stability properties. 5 Agai, the CTCS or leapfrog scheme is ) + U + c m+ U ), We ow seek a solutio of the form U m = U 0 expikm C). If C is real, this is a wave-like solutio. If C is complex, this solutio will behave expoetially, quite ulike the solutio of the cotiuous equatio. Substitutig U m ito the FDE, we fid that C = k si c Exercise: Verify this expressio for C. ) si k. Distortio of the Phase Speed We cosider the simple -D advectio equatio u t + c u x, where ux, t) depeds o both x ad t. The advectio speed c is costat ad, without loss of geerality, we assume c > 0. The equatio has a geeral solutio of the form u = fx ct), where f is a arbitrary fuctio. I particular, we may cosider the siusoidal solutio u = u0) expikx ct) of wavelegth L = π/k. We use cetered differece approximatios ) + U + c m+ U ), i both time ad space CTCS). Here U m = Um, ). Agai, C = k si ) c si k. If the argumet of the arcsie is less tha uity, C is real. Otherwise, C is complex, ad the solutio grows with time. Clearly, c/ is a sufficiet coditio for real C. It is also ecessary: for a wave of four gridlegths, we have k = π/, so that si k =. Thus, the coditio for stability of the solutio is Le c. This o-dimesioal parameter is ofte called the Courat umber, but is deoted here as Le for Lewy, who first discovered this stability criterio. 6

3 The above aalysis may be repeated for a implicit discretizatio six-poit Crak-Nicholso scheme): + + c U m+ + U m+ + U + ). The the phase speed C of the umerical solutio is C = k ta c ) si k. Exercise: Verify this result. Hit: Substitute U m = U 0 expikm C) ito the equatio. This equatio cotais a iverse taget term istead of the iverse sie occurrig i the leapfrog scheme. Thus, the umerical phase speed C is always real, so the scheme is ucoditioally stable. 9 Hits for MatLab Exercise. There are too may parameters. It is coveiet to reduce the umber by costructig o-dimesioal quatities. So, we defie κ = k ad µ = c. The the relatioships ca be writte: C c = κµ si µ si κ) for the explicit scheme. C c = κµ ta µ si κ) for the implicit scheme. Now there are oly two parameters, κ ad µ. You should plot curves of C/c as fuctios of µ for a selectio of values of κ, say for κ {0, 0 π, π 0,..., π}, with µ varyig from zero to, say, 0. It is easily show that C c ad that C π/k as c. Thus, the implicit scheme slows dow the faster waves. MatLab Exercise: Write a program to evaluate C = k si c ) si k. ad determie the behaviour of C i the limits c ad c. Write a program to evaluate C = k ta c ) si k. ad determie the behaviour of C i the limits c ad c. Exercise. Cosider the four-poit Crak-Nicholso scheme + + U m+ + U m+ + c U + m+ U + m + U m+ U m Show that the computatioal phase speed is give by C = ) c k ta ta k. Hit. Substitute U m = U 0 expikm C) ito the equatio. 0

4 Implicit Time Schemes I implicit schemes the advectio or diffusio terms are writte i terms of the ew time level variables. u PDE: t + c u x ) + FDE: + U + c α m+ ) + α) U + m+ U m+ ) U + m+ U m + For α =, this is the four-poit Crak-Nicholso scheme. The factor α determies the weight of the old time values compared with the ew time values i the FDE. ) Usig the vo Neuma method, we substitute U m = Aρ e imκ = Ae imκ θ) ito the FDE where κ = k ad θ = ω). Note that, for α = /, the scheme is cetered i time at U +/ m+/ which is ot at a gridpoit i space or i time). We multiply by e iκ/ ad obtai the amplificatio factor cos κ ρ = iµα si κ iµα ta κ cos κ + iµ α) si κ = + iµ α) ta κ Thus, the squared modulus of the amplificatio factor is ρ + 4µ α ta κ = + 4µ α) ta κ This implies ρ if α 0.5, i.e., if the ew values are give at least as much weight as the old values. 3 4 Agai, ρ + 4µ α ta κ = + 4µ α) ta κ For α 0.5, there is o restrictio o the size of! Absolute stability idepedet of the Courat umber is typical of implicit time schemes. I a implicit scheme, a poit at the ew time level is iflueced by all the values at the ew level, which avoids extrapolatio, ad is absolutely stable. Note also that if α < 0.5 the implicit time scheme reduces the amplitude of the solutio: it is a example of a dampig scheme. This property is useful for solvig problems such as spuriously growig moutai waves i semi-lagragia schemes. Schematic of a implicit scheme. Note that with the implicit scheme there is o extrapolatio.

5 Summary I If we cosider a marchig equatio du dt = F U) explicit methods such as the forward scheme U + U = F U ) or the leapfrog scheme are either Coditioally stable, or Absolutely ustable. U + U = F U ) Summary II A fully implicit scheme U + U = F U + ) ad a cetered implicit scheme U + U U + U + ) = F are absolutely stable. The latter scheme is attractive because it is cetered i time, ad it ca be writte with cetered space differeces, which makes it secod order i space ad i time. As these schemes have oly two time levels, they have o computatioal mode. Break here 7 The Crak-Nicholso Scheme The Crak-Nicholso Scheme is cetered i both time ad space. It has good stability ad accuracy properties. It is secod order accurate ad ucoditioally stable. Two forms of the Crak-Nicholso scheme for the advectio scheme are commoly used: The four-poit C-N scheme: + + U m+ + U m+ + c U + m+ U + m + U m+ U m The six-poit C-N scheme: + + c U + m+ U + + U m+ U 8

6 Domai of Depedece of Implicit Scheme * * * * * * + * * * * * * * m-3 m- m- m m+ m+ m+3 The lie of bullets ) represets a parcel trajectory. The value at the poit m at time + ) depeds o all the poits deoted by red asterisks *). Thus, the computatioal domai of depedece surrouds the physical domai of depedece. This is a ecessary coditio for a stable scheme. Implicit Schemes & Liear Systems All implicit schemes also have a sigificat disadvatage. Sice U + appears o the left- ad o the right-had sides, the solutio for U +, requires the solutio of a system of equatios. If it ivolves oly tridiagoal systems, this is ot a obstacle, because there are fast methods to solve them. There are also methods, such as fractioal steps with each spatial directio solved successively), where oe space dimesio is cosidered at a time. These so-called ADI alteratig directio implicit) schemes allow large time steps without a large additioal computatioal cost. Example of a Liear System. The -D advectio equatio o a periodic domai is u t + c u ul, t) = u0, t) x The implicit scheme six-poit Crak-Nicholso) scheme is + + c U + m+ U + + U m+ U We ca write this i matrix form with µ = c/4) +µ 0... µ U + 0 µ µ µ +µ... 0 U + 0 µ... 0 U + +µ µ = 0 +µ µ µ U + M where x M = M ad U M = U 0 for all. U 0 U U. U M Symbolically, the equatio may be writte The formal solutio of this is M U + = M U U + = M M U However, this requires the iversio of a M M matrix. There are much better ways to solve it. The matrix M is periodic tri-diagoal. There are very efficiet umerical methods of ivertig a system with such a matrix. The o-periodic problem, with U0 ad U M give, results i a slightly differet matrix, also tri-diagoal. If the o-liear terms are treated implicitly, we must solve a oliear algebraic system every time step. This is ormally impractical.

7 The possibility of usig a time step with a Courat umber much larger tha i a implicit scheme does ot guaratee that we will obtai accurate results ecoomically. The implicit scheme maitais stability by slowig dow the solutios, so that the waves satisfy the CFL coditio. We saw this clearly i the aalysis of the six-poit Crak- Nicholso scheme. For this reaso, implicit schemes are useful for those modes that are very fast but of little meteorological importace. We will ext cosider schemes i which the gravity wave terms are implicit while the remaiig terms are explicit. These semi-implicit schemes are of crucial importace i moder NWP. Exercise: See Notes ad Exercises, Kalay, pp Coclusio of

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