Computational Geodynamics: Advection equations and the art of numerical modeling

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1 540 Geodynamcs Unversty of Southern Calforna, Los Angeles Computatonal Geodynamcs: Advecton equatons and the art of numercal modelng Sofar we manly focussed on dffuson equaton n a non-movng doman. Ths s maybe relevant for the case of a dke ntruson or for a lthosphere whch remans undeformed. However there are also cases where materal moves. An eample s a plume rsng through a convectng mantle. The plume s hot and hence ts densty s low compared to the colder mantle around t. The hot materal rses wth a gven velocty (when term-projects are fnshed we wll have codes that gve us those veloctes for a gven densty dstrbuton). If the numercal grd remans fed n the background, the hot temperatures should be moved to dfferent grdponts (see fg. 1 for an llustraton of ths effect). Fgure 1: Snapshots of a bottom heated thermal convecton model wth a Raylegh-number of and constant vscosty (no nternal heatng). Temperature s advected through a fed (euleran) grd (crcles) wth a velocty (arrows) that s computed wth a Stokes solver. Mathematcally, the temperature equaton gets an addtonal term and n 1-D the dffuson-advecton equaton becomes (see equaton cheat sheet) ( ) ρc p t + v = ( k ) (1) or n 2-D ( ) ρc p t + v + v z = ( k ) + ( k ) (2) z z z where v,v z are veloctes n -,respectvely z-drecton. If dffuson s absent (.e. k = 0), the advecton equatons are t + v = 0 (3) and t + v + v z z = 0 (4) These equatons should also be solved f you want to compute the settlng of a heavy partcle n a flud, or n weather predcton codes. In the current lecture we re gong to look at some optons on how to solve these equatons wth a fnte dfference scheme on a fed grd. Even though the equatons are farly smple, t s far from smple to solve them accurately. If not done carefully, you ll up wth strong numercal artfacts such as wggles and numercal dffuson.

2 % advecton_central_1d % clear all n = 201; W = 40; % wdth of doman Vel = -4; % velocty sgma = 1; Ampl = 2; nt = 500; % number of tmesteps dt = 1e-2; d = W/(n-1); = 0:d:W; % Intal gaussan T-profle c = 20; T = Ampl*ep(-(-c).ˆ2/sgmaˆ2); % Velocty V = ones(1,n)*vel; % Central fnte dfference dscretzaton for tme=1:nt % central fn. dff for =2:n-1 Tnew() =?? Tnew(1) = T(1); Tnew(n) = T(n); T = Tnew; tme = tme*dt; % Analytcal soluton for ths case T_anal = Ampl*ep(-(-c-tme*Vel).ˆ2/sgmaˆ2); fgure(1),clf, plot(,t,,t_anal), drawnow leg( Numercal, Analytcal ) Fgure 2: MATLAB scrpt to be used wth eercse 1. FTCS method In a 1-D case the smplest way to dscretze equaton 3 s by employng a central fnte dfference scheme (also called the forward-tme, central space or FTCS scheme): T n+1 T n = v, T n +1 T n 1 2 (5) Eercse 1 Program the FTCS method n the code of fgure 2 and watch what happens. Change the sgn of the velocty. Change the tmestep and grdspacng and compute the nondmensonal parameter α = v /. When do unstable results occur? As you saw from the eercse, the FTCS method does not work... In fact t s a nce eample of a scheme that looks nce (and logcal!) on paper, but sucks [you can actually mathematcally prove ths by a socalled von Neuman stablty analyss, f you re nterested you should read chapter 5 of Marc Spegelman s scrpt on the course webpage; t s related to the fact that ths scheme produces ant-dffuson, whch s numercally nstable]. La method Some day, long ago, there was a guy called La and he thought: well let s replace the T n n the tme-dervatve of equaton 5 wth (T+1 n + T 1 n )/2. The resultng equaton s T n+1 (T+1 n + T 1 n )/2 T+1 n = v T n, 2 1 (6) 2

3 Eercse 2 Program the La method by modfyng the scrpt of the last eercse. Play wth dfferent veloctes and compute the Courant number α, whch s gven by the followng equaton: α = v (7) Is the numercal scheme stable for all Courant numbers? As you saw from the eercse the La method doesn t blow up, but does have a lot of numercal dffuson. So t s an mprovement but t s stll not perfect, snce you may loose the plumes of fgure 1 around md-mantle purely due to numercal dffuson (and than you ll probably fnd some fancy geochemcal eplanaton for ths...). I hope that you also slowly start to see that numercal modelng s a bt of an art... Upwnd scheme The net scheme that people came up wth s the so-called upwnd scheme. Here, the spatal fnte dfference scheme deps on the sgn of the velocty: T n+1 T n = v, { T n T n 1, f v, > 0 T+1 n T n, f v, < 0 (8) Eercse 3 Program the upwnd scheme method. Play wth dfferent veloctes and compute the Courant number α Is the numercal scheme stable for all Courant numbers? Also the upwnd scheme suffers from numercal dffuson. Sofar we employed eplct dscretzatons. You re probably wonderng whether mplct dscretzatons wll save us agan ths tme. Bad news: doesn t work (try f you want). So we have to come up wth somethng else. Staggered Leapfrog The eplct dscretzatons dscussed sofar were second order accurate n tme, but only frst order n space. We can also come up wth a scheme that s second order n tme and space T n+1 T n 1 T+1 n = v T n, 2 2 the dffculty n ths scheme s that we ll have to store two tmestep levels: both T n 1 and T n (but m sure you ll manage). Eercse 4 Program the staggered leapfrog method (assume that at the frst tmestep T n 1 = T n ). Play around wth dfferent values of the Courant number α. Also make the wdth of the gaussan curve smaller. The staggered leapfrog method works qute well regardng the ampltude and wavespeed as long as α s close to one. If however α << 1 and the lengthscale of the to-be-transported quantty s small compared to the number of grdponts (e.g. we have a thn plume), numercal oscllatons agan occur. Ths s typcally the case n mantle convecton smulatons (see Fg. 1), so agan we have a numercal scheme that s not great for mantle convecton smulatons. 1 (9) 3

4 MPDATA Ths s a technque that s frequently appled n (older) mantle convecton codes. The basc dea s based on an dea of Smolarkewcz and s bascally a modfcaton of the upwnd scheme, where some teratve correctons are appled. The results are qute ok. However t s somewhat more complcated to mplement (the system that s solved s actually nonlnear and requres teratons, whch we dd not dscuss yet). Moreover we stll have a restrcton on the tmestep (gven by the Courant crterum). For ths reason we won t go nto detal here (see Spegelman s lecture notes, chapter 5 f you re nterested). Sem-Lagrangan So what we want s a scheme that doesn t blow up, has only small numercal dffuson and that s not lmted by the Courant crterum. It actually ests and s called the sem-lagrangan method. I m a bg fan of t and so s Marc Spegelman (see hs course notes), and people that do clmate modellng or numercal weather predcton. It s however a bt werd and has lttle to do wth the fnte dfference schemes we dscussed sofar. Snce ths scheme s however the one that s maybe most mportant for practcal purposes we wll go n more detal here. Basc dea The basc dea of the sem-lagrangan method s llustrated n fgure 3A and s gven by the followng For each pont 1. Assume that the future velocty v (n + 1,) s known. Under the assumpton that the velocty at the old tmestep s almost dentcal to the future velocty (v (n+1,) = v (n,)) and that velocty doesn t vary spatally (v (n, 1) = v (n,) = v (n,+1)), we can compute the locaton X where the partcle came from by X= () v (n+ 1,) 2. Interpolate your temperature from grdponts to the locaton X at tme n. Use cubc nterpolaton (n MATLAB use the command nterp1(,t, pont, cubc ) for nterpolaton). Now you smply say that T (n+1,) = T (n,x), and you re done. Note that ths scheme assumes that no heat-sources were actve durng the advecton of T from T (n,x) to T (n + 1,). If heat sources are present and are spatally varable, some etra stuff needs to be done (read Marc s lecture notes). Eercse 5 Program the sem-lagrangan advecton scheme llustrated n fgure 3A. Is there a Courant crterum for stablty? Some mprovements The algorthm descrbed n fgure 3A llustrates the basc dea of the sem-lagrangan scheme. However t has two problems. Frst t assumes that velocty s spatally constant (whch s clearly not the case n mantle convecton smulatons). Second, t assumes that velocty doesn t change between tme n and n + 1. We can overcome both problems by usng a more accurate tmesteppng algorthm. An eample s an teratve md-pont scheme whch works as follows (see fgure 3B): For each pont 1. Use the velocty v (n + 1,) to compute the locaton at tme n + 1/2 (n other words: take half a tmestep backwards n tme). 2. Fnd the velocty at the locaton at half tmestep n + 1/2. Assume that the velocty at the half-tmestep can be computed as v (n + 1/2,) = v (n+1,)+v (n,) 2. Use lnear nterpolaton for the spatal nterpolaton of velocty v (n + 1/2,). 4

5 A tme t n+1 n T(n,-2) v (n,-1) T(n,-1) v (n+1,) v (n,) T(n,) B tme t n+1 n+1/2 n v (n,-2) true path v (n,-1) v (n+1,) v (n,) space space Fgure 3: Bascs of the sem-lagrangan method. See tet for eplanaton. 3. Go back to pont 1, but use the velocty v (n + 1/2,) nstead of v (n + 1,) to move the pont (n + 1,) backwards n tme. Repeat ths process a number of tmes (for eample 5 tmes). Ths gves a farly accurate centered velocty. 4. Compute the locaton X wth the centered velocty v (n + 1/2,), wth X= () v (n + 1/2,). 5. Use cubc nterpolaton to fnd the temperature at pont X. Other methods are also possble. A popular one s the fourth order Runga-Kutta method, whch s mplemented smlarly (but takes a bt more work). Eercse 6 Program the sem-lagrangan advecton scheme wth the centered mdpont method as llustrated n fgure 3B. If you don t understand the recpe above check Marc s lecture notes, page 67 for a MATLAB pseudocode. Some care has to be taken f pont X s outsde of the computatonal doman, snce MATLAB wll return NaN for the velocty (or temperature) of ths pont. Use the velocty v (n + 1,) n ths case. A pseudocode s gven by f snan(velocty) Velocty = V() 2D advecton Snce I m convnced that the sem-lagrangan method s the only good advecton algorthm (ecept n the case of pseudospectral methods), ths s the one we re gong to mplement n 2D. Assume that velocty s gven by v (,z) = z (10) v z (,z) = (11) Moreover, assume that the ntal temperature dstrbuton s gaussan and gven by (( ( ) 2 + z 2) ) T (,z) = 2ep (12) wth [ 0.5,0.5], z [ 0.5,0.5]. Eercse 7 Program advecton n 2D usng the sem-lagrangan advecton scheme wth the centered mdpont method. Use the MATLAB routne nterp2 for nterpolaton and employ lnear nterpolaton for velocty and cubc nterpolaton for temperature. A MATLAB scrpt that ll get you started s shown on fgure 4. 5

6 % sem_lagrangan_2d: 2D sem-lagrangan wth center mdpont tme steppng method % clear all W = 40; % wdth of doman sgma =.1; Ampl = 2; nt = 500; % number of tmesteps dt = 5e-1; % Intal grd and velocty [,z] = meshgrd(-0.5:.025:0.5,-0.5:.025:0.5); nz = sze(,1); n = sze(,2); % Intal gaussan T-profle T = Ampl*ep(-((+0.25).ˆ2+z.ˆ2)/sgmaˆ2); % Velocty V = z; Vz = -; for tme=1:nt V_n = V; % Velocty at tme=n V_n1 = V; % Velocty at tme=n+1 % V_n1_2 =??; % Velocty at tme=n+1/2 % Vz_n =??; % Velocty at tme=n % Vz_n1 =??; % Velocty at tme=n+1 % Vz_n1_2 =??; % Velocty at tme=n+1/2 Tnew = zeros(sze(t)); for =2:n-1 for z=2:nz-1 V_cen = V(z,); Vz_cen = Vz(z,); % for?? % X =? % Z =? %lnear nterpolaton of velocty % V_cen = nterp2(,z,?,?,?, lnear ); % Vz_cen = nterp2(,z,?,?,?, lnear ); f snan(v_cen) V_cen = V(z,); f snan(vz_cen) Vz_cen = Vz(z,); % % X =?; % Z =?; % Interpolate temperature on X % T_X = nterp2(,z,?,?,?, cubc ); f snan(t_x) T_X = T(z,); Tnew(z,) = T_X; Tnew(1,:) = T(1,:); Tnew(n,:) = T(n,:); Tnew(:,1) = T(:,1); Tnew(:,n) = T(:,n); T = Tnew; tme = tme*dt; fgure(1),clf pcolor(,z,t), shadng nterp, hold on, colorbar contour(,z,t,[.1:.1:2], k ), hold on, quver(,z,v,vz, w ) as equal, as tght drawnow pause Fgure 4: MATLAB scrpt to be used wth eercse 7. 6

7 Advecton and dffuson: operator splttng Sofar we learned that the sem-lagrangan methods are probably the best methods for advecton domnated problems. We also learned how to solve the dffuson equaton n 1-D and 2-D. In geodynamcs, we often want to solve the coupled advecton-dffuson equaton, whch s gven by equaton 1 n 1-D and by equaton 2 n 2-D. We can solve ths pretty easly by takng the equaton apart and by computng the advecton part separately from the dffuson part. Ths s called operator-splttng, and what s done n 1-D s the followng. Frst solve the advecton equaton T n+1 T n + v = 0 (13) for eample by usng a sem-lagrangan advecton scheme. Then solve the dffuson equaton ρc p T t = ( k ) T + Q (14) and you re done (note that I assumed that Q s spatally constant; f not you should consder to slghtly mprove the advecton scheme by ntroducng source terms). In 2D the scheme s dentcal (but you ll have to fnd your old 2D codes...). Eercse 8 Program dffuson-advecton n 2D usng the sem-lagrangan advecton scheme coupled wth an mplct 2D dffuson code (from last week s eercse). Base your code on the scrpt of fgure 4. 7