Taylor epasio: Let ƒ() be a ifiitely differetiable real fuctio. A ay poit i the eighbourhood of 0, the fuctio ƒ() ca be represeted by a power series of the followig form: X 0 f(a) f() f() ( ) f( ) ( ) ( ) ( ) () f ( 0) f( 0) f ( 0) f ( 0) 0 0 + 0 + 0 + 0 0!!!! where! stads for the factorial of ad f () ( 0 ) for the -derivative of f at 0. Eamples: The fuctio f() e i the eighborhood of 0 has the followig Taylor series: 4 e + + + + 6 4 Show that the TE of f() si() aroud si() - +! 5! Q: 5 0 is: e What is the Taylor epasio (-TE) of f() aroud 0? si( ) e Let has a epasio of the form: si( ) e c +c +c +c si( ) 5 e c +c +c +c - +! 5! S.
Iterpolatio techiques By iterpolatio we mea a procedure for estimatig the fuctio at itermediate values. If for < < the fuctios f ( ) ad f ( ) are used to evaluate f (), the the procedure is called iterpolatio. By etrapolatio we mea a procedure for estimatig the fuctio f() at a eterior ew value: If for < < or < < the fuctios f ( ) ad f ( ) are used to evaluate f (), the the procedure is called EXTRA-polatio. f(x ) f(x ) F? f(x ) f(x ) F? Liear iterpolatio: y( ) - y( ) y() - y( ) y( ) - y( ) y() - y( ) ( ) y( ) y( ) + y() - y( ) ( ) S.
Polyomial iterpolatio: Assume we are give the followig poits: (,y ), (,y ),, (,y ). 0 0 Usig these poits, we wat to have a approimatio for y at. The procedure is to costruct a polyomial of degree, such that y() a + a +... a. - - 0 This polyomial goes through the above + poits, therefore is satisfies the followig equatios: a + a +... a 0-0 0 0 0 0 0-0 a + a +... a y a + a +... a y - 0 y a y a y a0 y `X 0 Q: what would it mea, if the matri X turs to be sigular? Q : Assume we are give the followig poits: (,y ) (,) Use these poits to fid a 0 0 (,y ) (,) polyomial approimatio for (,y ) (,) y at 5/? S.
Lagrage iterpolatio: Assume we are give the followig poits: (,y ), (,y ),, (,y ). 0 0 Based o these poits, Lagrage suggested the followig approimatio for y at : N y y( ) y() y y (), where - 0 () - k Eample Give are the followig set of poits (,y): 4 y 8 7 6 5 - - - 0() ( 0-0- 0-6 - - - k 0 () ( )( )( 4) - 0 - - - - - 0 () ( )( )( 4) - 0 - - k )( )( 4) - 0 - - () ( )( )( ) - 0 - - 6 y() 8 ( )( )( 4) 7 ( )( )( 4) 6 + + 6 ( )( )( 4) + 5 ( )( )( ) 6 S.4
Fiite differece discretizatio: L 7 & 8: MHD/ZAH How to represet u T i fiite space? Forward differece (u < 0): T ΔT T( ) - T( ) T - T Δ - h fiite space + + fiite space - - T ΔT T( ) - T( ) T - T Δ - h - Ad how good is this approatio? Taylor epasio T - T( -) T( -h) T h T + T + O(h ) Subtitute this epressio: + But how good is this approatio? Taylor epasio T + T( +) T( + h) T + h T + T + O(h ) Subtitutig this epressio: ΔT T + - T T + ht +(h /) T T T + O(h) Δ h h The scheme is first order accurate. Backward differece (u > 0): h h Trucatio error ΔT T - T- T (T- ht +(h /) T ) T + O(h) Δ h h The scheme is first order accurate also. Trucatio error S.5
Higher order derivatives: L 7 & 8: MHD/ZAH T T ( ) F ΔF F( ) - F( ) F - F F Δ - h F fiite space - + ΔT - T - T ΔT T - T - +, F+ Δ h Δ h + Subtitute these epressio: T ΔF F -F- F - F - T + - T T - Δ h h h + But how good is this approatio? Taylor epasio T ± T( ± ) T( ± h) T ± h T + T + O(h ) h Δ ΔT ( ) T + - T + T - T + Δ Δ h O(h ) The scheme is secod order accurate. S.6
The oe-dimesioal heat equatio: Cosider the followig heat diffusio equatio: I the absece of trasport, the heat diffuses i a symmetric maer, provided the diffusio coefficiet, χ, ad the BC are symmetric too. If advectio (trasport) is icluded, the this symmetry will be broke as there is a preferable directio for heat trasport, which is eemplified i the followig two figures. T t T χ. Assume that both edges of the metal rod are kept at certai costat temperatures T u ad T d. Let the rod be heated at the ceter for a certai period of time. How do the iitial, itermediate ad fial profiles of the temperature look like? Ituitively (without performig aalytical or umerical calculatio) we may epect that: the BCs ad the ICs play a essetial role i determiig the form ad evolutio of the solutio at ay time t. There are two types of problems: Iitial value ad boudary value problems. The stregth of depedece o the ICs ad BCs determie the type of the problem. S.7
The heat equatio: discretizatio: The temperature depeds o t as well as o, i.e, T T(t,). Thus for each value of ad value of t there is a suitable value for T (hopefully a uique value). T t T χ T T(t, ) T t t A eplicit discretizatio of this equatio yields the followig form: + T T χ T+ T T T + T ( ) f T δt + (poitwise) T st + ( s) T + st+, Time Eplicit δt χ where s discretizatio S.8
T t T χ T T t t L a (T) χ + O( δt) + O( ) Trucatio error T χ T is Cosistecy: The fiite space represetatio La ( T ) of the equatio t said to be cosistet, if the trucatio error goes to zero as δt, 0. (local aalysis) T χ T is Stability: The fiite space represetatio La ( T ) of the equatio t said to be umerically stable, if accumulated errors do ot grow with time (o-local). S.9
Stability aalysis: Assume we are give the followig solutio procedure i the discretizatio space: T st + ( s) T + st, where s χδ t / + + The computer provides ot T + but T T st + ( s) T + st + * * * + * Defie: ξ T T +, which is abtaied from: ξ sξ + ( s) ξ + sξ, + + Takig itp accout that ξ BC 0, the: + ξ ( s) ξ + sξ + ξ sξ + ( s) ξ + sξ4 + ξ + ξ sξ + ( s) ξ + sξ+ + ξj sξj + ( s) ξ J s s s s s where A s s s Aξ, Matri algebra tells us that ξ is bouded, if the absolute value of the maimum eigevalue of A: λm. S.0
The eigevalues of A are: Therefore, we require: L 7 & 8: MHD/ZAH mπ λ m 4s si ( ), m,,... J- ( J ) mπ 4s si ( ) ( J ) The RHS is straightforwardly satisfied, BUT the LHS requires: mπ s si ( ) ( J ) δ t s. Defiig C δ t/, we coclude that the methos is stable for C, which is the so called courat-friedrich-levy umber. Covergece: The fiite space represetatio La ( T ) of the equatio t said to coverge, if it is cosistet ad umerically stable. This is a cosequece of La s equivalece theorem: La's equivalece theorem: ``Give a properly Cosistecy + stability covergece posed iitial value problem ad a fiite-differece. T χ T is S.
Weak ad strog solutios of Navier-Stokes equatios The equatios describig the evolutio of icompressible viscous fluid flows are called the Navier-Stokes equatios (that were formulated aroud 80s) ad read: where u u u ν (u,u y,u z), t, viscosity, p pressure, Ω domai i R, Γ boudary, I (0, T) ad u 0 iitial value. For a give data, it was prove by Leray (94) that the Navier-Stokes equatios have at least oe weak solutio. However, it is still ot clear, whether the weak solutio is uique or ot. Now by a weak solutio we mea a solutio that satisfies the above partial differetial equatios o average, but ot ecessary i a poitwise maer. For eample, the derivatives may ot eist at certai poits. However, a strog solutio is said to satisfy the equatios everywhere ad at each poit of the domai La-Wedroff theorem: The umerical solutio q is said to be a weak solutio for the aalytical equatio Lq ( ), if the umerical scheme employed is coservative, cosistet ad the umerical procedure coverges. Coservatio + cosistecy + covergece weak solutio La theorem: If q has bee obtaied usig a coservative ad cosistet scheme, the q coverges to a weak solutio of the aalytical problem whe, i.e., whe the umber of grid poits goes to ifiity. For umerical mathematicias, the above-metioed theorems implies the followig: Differet coservative umerical methods may yield differet solutios, provided the umber of grid poits is relatively small. Differet coservative umerical methods that are umerically stable ad cosistet must coverge to the same weak solutio if the umber of grid poits goes to ifiity. You caot claim to have foud a solutio for the physical problem, uless you carried out the caculatios with sufficietly large umber of grid poits, beyod which doublig the umber of grid poits yield o oticeable improvemet. From Fletcher : Computatioal techiques... (990) S.
From Fletcher : Computatioal techiques... (990) S.
T T Q : Give is the heat equatio: χ i the domai D[t] [] [0,] [0,] together with t the IC ad BC: T(t0), T(t,0), T(t,). + Solve the equatio usig the FTCS formulatio, i.e., T st + ( s) T + st, where followig s-values: s 0., 0., 0.4, 0.8,.., + s χδ t /, χ ad N ( umber of grid poits i -directio) 00 for the Plot the solutios at times: t 0., 0., 0.4 ad.0. S.4