The heat equation. But how to evaluate: F F? T T. We first transform this analytical equation into the corresponding form in finite-space:
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1 L9_0/Cop. Astro.-, HS 0, Uiversität Basel/AAH The heat equatio T t T x. We first trasfor this aalytical equatio ito the correspodig for i fiite-space:.. t T δt T T F F F F F x x x x T T T F x But how to evaluate: F F? F T T T T T T, F x x T T T F F T T T T T oldew old, ew,,? T T T The operators o the RHS ca be evaluated fro differet tie levels, e.g., explicit (), iplicit () or ea values of the I the case of iplicit discretizatio: T T T T T T T T T T st st st, where s st ( s) T st T Let s, the T 3T T T These equatios ca be expaded for, 3,...Jas follows: 3... J 3 T T T T BC 3 T 3T T T T 3T T T BC J J J J L9_0/S.
2 L9_0/Cop. Astro.-, HS 0, Uiversität Basel/AAH The idividual equatio at each grid ca be put together to for a atrix: 3 T T T 3 T3 T3 T T 3 TJ TJ TJ A T RHS For solvig this equatio, we ay costruct a Thoas-Algorith or use available ad optiized routies, such as the GTSL-routie: The Tri-diagoal solver: GTSL CALL GTSL(Nel,Sub,Diag,Super,RHS,INFO) where Nel total uber of diagoals Sub sub-diagoal etries Diag diagoal etries Super super-diagoal etries RHS the etries o the right had side INFO 0, if all etries are defied k, if etry uber k is ot appropriately defied L9_0/S.
3 L9_0/Cop. Astro.-, HS 0, Uiversität Basel/AAH The Crak-Nicholso Method (CN): The CN ethod relies o evaluatig the RHS, usig old ad ew values, such as half old half ew, i.e., soe sort of a ea value. The advatage is that the ethod is ore accurate i tie (secod order accurate i tie which eas that errors are of the order of, however, it requires ore arithetic operatios ad eory. The ethod operates as follows: T T oldew T T T T T T T T T T T [ old ] [ ew] T T T T T T T T T T st ( s) T st ( ) RHS T st st st, where s. st ( s) T st st ( s) T st RHS st s T st s s T RHS st s s s T3 RHS3 s s s TJ RHSJ- s s T J RHS J- st J L9_0/S.3
4 L9_0/Cop. Astro.-, HS 0, Uiversität Basel/AAH The advectio-diffusio heat equatio: T T Advectio T VT T., t x t x x Advectio ter where V f(x), for exaple: V. Upwid discretizatio: the idea here is that the atter/eergy i a test volue V is iflueced aily by the flux coig fro the up-strea directio rather tha fro the dow-strea. The upwid diskretizatio procedure: VT x - VT -V T Upwid V [ α T ( - α )T ] - V [ α T (- α )T ] α V ( α )V α V ( α )V T T - T - x x x x CALL SUPWIND(T,V, α) : IF(V <0) THEN α () ELSE α () 0 ENDIF L9_0/S.4
5 L9_0/Cop. Astro.-, HS 0, Uiversität Basel/AAH I the explicit case, the diffusio ad advectio operators are evaluated, usig the values fro the old tie level. The procedure rus as follows: Explicit calculatio: T T T T T -T V T -V T T -T V TUP -VTUP T T T T - T st ( ) s T st -β TUP - β TUP, s x where ad UPWIND(T,V,TUP) : V β IF(V <0) THEN ELSE ENDIF TUP() T TUP() T Q: Costruct a solver for solvig the followig equatio explicitly: T VT T, i the iterval [ x ] subect to t x x the coditios: T(t, x), T(t, x) ad V cost. T(t0, x). Show the resultig profiles of T at t0.5, 0.5, 0.75 ad.0 L9_0/S.5
6 L9_0/Cop. Astro.-, HS 0, Uiversität Basel/AAH I the iplicit case, the diffusio ad advectio operators are evaluated, usig the values fro the NEW tie level. The procedure rus as follows: Iplicit calculatio: T -T V T -V T T T T T T -T V [ αt - (- α)t ] - V [ αt (- α)t ] T T T T T -T α V ( α )V α V ( α )V T - T T T T T T S T D T S T T Q: Copute: S, D, S. Q3: Prove that upwid-discretizatio boosts the diagoal doiace of the coefficiet atrix. Q4 : Costruct a solver for solvig the followig equatio iplicitly: T VT T, t x x i the iterval [ x ] subect to the coditios: T(t, x), T(t, x) ad V cost. T(t0, x). Show the profiles of T at t0.5,.0, 0.0. L9_0/S.6
7 L9_0/Cop. Astro.-, HS 0, Uiversität Basel/AAH We ay cobie explicit ad iplicit tie steppig to geerate the daped CN-ethod as follows: Crak-Nicholso ethod for advectio-diffusio equatios: T -T V T -V V T -V T T α CN (- αcn ) T T T T αc N We defie the paraeter αcn, to be 0 αcn. T T T T ( -αcn ) T -T V T -V T T T T T α CN - V T -V T T T T T (- αcn ) S T - D T - S T - T V T -V T T T T T (-αc N ) Q5: Costruct a Crak-Nicholso solver for solvig the equatio: T VT T, i the iterval [ x ] subect to t x x the coditios: T(t, x), T(t, x) ad V cost. T(t0, x). Show the profiles of T at t0.5, 0.75,.0, 0.0 usig α CN δ t. δ t L9_0/S.7
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