2 Finite difference basics

Transcription

1 Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one- The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T ρc p t = k T ) where ρ s densty, c p heat capacty, k thermal conductvty, T temperature, x dstance and t tme. If the thermal conductvty, densty and heat capacty are constant over the model doman, the equaton can be smplfed to T t = T κ2 2 2) where κ = k ρc p s the thermal dffusvty a common value for rocks s κ = 10 6 m 2 s 1 ). We are nterested n the temperature evoluton versus tme T x, t) whch satsfes equaton 2, gven an ntal temperature dstrbuton Fg. 1A). An example would be the ntruson of a basaltc dke n cooler country rocks. How long does t take to cool the dke to a certan temperature? What s the maxmum temperature that the country rock experences? The frst step n the fnte dfferences method s to construct a grd wth ponts on whch we are nterested n solvng the equaton ths s called dscretzaton, see Fg. 1B). The next step s to replace the contnuous dervatves of equaton 2 wth ther fnte dfference approxmatons. The dervatve of temperature versus tme T t can be approxmated wth a forward fnte dfference approxmaton as T t T n+1 T n t n+1 t n = T n+1 T n t 1) = T new T current t here n represents the temperature at the current tmestep whereas n + 1 represents the new future) temperature. The subscrpt refers to the locaton Fg. 1B). Both n and are ntegers; n vares from 1 to n t total number of tme steps) and vares from 1 to n x total number of grd ponts n x-drecton). The spatal dervatve of equaton 2 s replaced by a central fnte dfference approxmaton,.e., 2 T 2 = ) T T n +1 T n T n T n 1 3) = T n +1 2T n + T n 1 2 4) Substtutng equaton 4 and 3 nto equaton 2 gves T n+1 T n T n = κ +1 2T n + T n ) 1 t 2 5) The thrd and last step s a rearrangement of the dscretzed equaton, so that all known quanttes.e. temperature at tme n) are on the rght hand sde and the unknown quanttes on the left-hand sde propertes at n + 1). Ths results n: T T n+1 = T n n + κ t +1 2T n + T 1 n ) 2 6) Because the temperature at the current tmestep n) s known, we can use equaton 6 to compute the new temperature. The last step s to specfy the ntal and the boundary condtons. If for example the country rock has a temperature of 300 C and the dke a wdth of 2 meters, wth a magma temperature of 1200 C, we can wrte as ntal condtons: T x < 1, x > 1, t = 0) = 300 7) T 1 x 1, t = 0) = ) 1

2 Numersche Methoden 1, WS 11/12 B.J.P. Kaus A country rock dke B boundary nodes,n+1 L -1,n,n +1,n Tx,0) tme t,n-1 x x space L Fgure 1: A) Setup of the model consdered here. A hot basaltc dke ntrudes cooler country rocks. Only varatons n x-drecton are consdered; propertes n the other drectons are assumed to be constant. The ntal temperature dstrbuton T x, 0) has a step-lke perturbaton. B) Fnte dfference dscretzaton of the 1D heat equaton. The fnte dfference method approxmates the temperature at gven grd ponts, wth spacng. The tme-evoluton s also computed at gven tmes wth tmestep t. In addton we assume that the temperature far away from the dke center at L/2 ) remans at a constant temperature. The boundary condtons are thus T x = L/2, t) = 300 9) T x = L/2, t) = ) So now you have a feelng about fnte dfferences. The attached MATLAB code shows an example n whch the grd s ntalzed, and a tme loop s performed. In the exercse, you wll fll n the questonmarks and obtan a workng code that solves equaton 2. 3 Exercses 1. Open the MATLAB edtor and create an empty fle wth the name heat1dexplct.m. Fll n the queston marks and run the fle by typng heat1dexplct n the MATLAB command wndow make sure you re n the correct drectory). 2. Vary the parameters e.g. use more grdponts, a larger tmestep). Note that f the tmestep s ncreased beyond a certan value what does ths value depend on?), the numercal method becomes unstable. Ths s a major drawback of explct fnte dfference codes such as the one presented here. In the next lesson we wll learn methods that do not have these lmtatons. 3. Go through the rest of the handout and see how one derves fnte dfference approxmatons. 4. Record and plot the temperature evoluton versus tme at a dstance of 5 meter from the dke/country rock contact. What s the maxmum temperature the country rock experences at ths locaton and when s t reached? Assume that the country rock was composed of shales, and that those shales were transformed to hornfels above a temperature of 600 C. What s the wdth of the metamorphc aureole? 5. Bonus queston: Derve a fnte-dfference approxmaton for varable k and varable. 2

3 Numersche Methoden 1, WS 11/12 B.J.P. Kaus %heat1dexplct.m % % Solves the 1D heat equaton wth an explct fnte dfference scheme clear %Physcal parameters L = 100; % Length of modeled doman [m] Tmagma = 1200; % Temperature of magma [C] Trock = 300; % Temperature of country rock [C] kappa = 1e-6; % Thermal dffusvty of rock [m2/s] W = 5; % Wdth of dke [m] day = 3600*24; % # seconds per day dt = 1*day; % Tmestep [s] % Numercal parameters nx = 201; % Number of grdponts n x-drecton nt = 500; % Number of tmesteps to compute dx = L/nx-1); % Spacng of grd x = -L/2:dx:L/2;% Grd % Setup ntal temperature profle T = onesszex))*trock; Tfndabsx)<=W/2)) = Tmagma; tme = 0; for n=1:nt % Tmestep loop % Compute new temperature Tnew = zeros1,nx); for =2:nx-1 Tnew) = T) +?????; end % Set boundary condtons Tnew1) = T1); Tnewnx) = Tnx); % Update temperature and tme T = Tnew; tme = tme+dt; % Plot soluton fgure1), clf plotx,tnew); xlabel x [m] ) ylabel Temperature [^oc] ) ttle[ Temperature evoluton after,num2strtme/day), days ]) end drawnow Fgure 2: MATLAB scrpt to solve equaton 2 once the queston marks are flled...). 3

4 Numersche Methoden 1, WS 11/12 B.J.P. Kaus Taylor-seres expansons and fnte dfferences Fnte dfference approxmatons can be derved through the use of Taylor seres expansons. Suppose we have a functon fx), whch s contnuous and dfferentable over the range of nterest. Let s also assume we know the value fx 0 ) and all the dervatves at x = x 0. The forward Taylor-seres expanson for fx 0 + ) about x 0 gves fx 0 +) = fx 0 )+ fx 0) + 2 fx 0 ) ) fx 0 ) ) 3 2! 3 3! We can compute the frst dervatve by rearrangng equaton 11 fx 0 ) = fx 0 + ) fx 0 ) Ths can also be wrtten n dscretzed notaton as: fx ) 2 fx 0 ) ) 2 3 fx 0 ) ) 2 2! 3 3! = f +1 f + n fx 0 ) ) n n +O) n+1 11) n!... 12) + O) 13) here O) s called the truncaton error, whch means that f the dstance s made smaller and smaller, the numercal approxmaton) error decreases as. Ths dervatve s also called frst order accurate. We can also expand the Taylor seres backward fx 0 ) = fx 0 ) fx 0) + 2 fx 0 ) ) fx 0 ) ) 3 2! ) 3! In ths case, the frst backward) dervatve can be wrtten as fx 0 ) = fx 0) fx 0 ) + 2 fx 0 ) ) 2 3 fx 0 ) ) 2 2! ! 15) fx ) = f f 1 + O) 16) By addng equatons 12 and 15 and dvdng by two a second order accurate frst order dervatve s obtaned fx ) = f +1 f 1 + O) 2 17) 2 By addng equatons 11 and 14 an approxmaton of the second dervatve s obtaned f 2 x ) 2 = f +1 2f + f 1 ) 2 + O) 2 18) Wth ths approach we can bascally derve all possble fnte dfference approxmatons. A dfferent way to derve the second dervatve s by computng the frst dervatve at +1/2 and at 1/2 and computng the second dervatve at by usng those two frst dervatves: f 2 x ) 2 = fx +1/2 ) fx 1/2) x +1/2 x 1/2 = fx +1/2 ) fx 1/2 ) = f +1 f x +1 x 19) = f f 1 x x 1 20) f +1 f x +1 x f f 1 x x 1 0.5x +1 x 1 ) Smlarly we can derve hgher order dervatves. Note that the hghest order dervatve that usually occurs n geodynamcs s the 4 th -order dervatve. 21) 4

5 Numersche Methoden 1, WS 11/12 B.J.P. Kaus Fnte dfference approxmatons The followng equatons are common fnte dfference approxmatons of dervatves. If you n the future need to wrte a fnte dfference approxmaton, come back here. Left-sded frst dervatve, frst order u = u u 1 + O) 22) 1/2 Rght-sded frst dervatve, frst order u = u +1 u + O) 23) +1/2 Central frst dervatve, second order u = u +1 u 1 + O) 2 24) 2 Central frst dervatve, fourth order u = u u +1 8u 1 + u 2 + O) 4 25) 12 Central second dervatve, second order 2 u 2 = u +1 2u + u O) 2 26) Central second dervatve, fourth order 2 u 2 = u u +1 30u + 16u 1 u O) 4 27) Central thrd dervatve, second order 3 u 3 = u +2 2u u 1 u O) 2 28) Central thrd dervatve, fourth order 3 u 3 = u u +2 13u u 1 8u 2 + u O) 4 29) Central fourth dervatve 4 u 4 = u +2 4u u 4u 1 + u O) 2 30) Note that the hgher the order of the fnte dfference scheme, the more adjacent ponts are requred. It s also mportant to note that dervatves of the followng form k u ) 31) should be formed as follows k u ) = k +1/2 u+1 u u k u 1 1/2 + O) 2 32) 5

6 Numersche Methoden 1, WS 11/12 B.J.P. Kaus If k s spatally varyng, the followng approxmatons are wrong but are commonly made mstakes...)!!!! k u ) = k +1 u+1 u k u u 1 33) k u ) = k u +1 2u + u ) 6