Internatona Conerence on Advanced ectronc Scence and Technoogy (AST 6) A nte derence method or heat equaton n the unbounded doman a Quan Zheng and Xn Zhao Coege o Scence North Chna nversty o Technoogy Bejng 44 Chna Abstract. The numerca method o the one-dmensona non-homogeneous heat equaton on an unbounded doman s consdered. Two exact artca boundary condtons are apped on two artca boundares to mt the orgna probem onto a bounded computatona doman. Then the nte derence method s deveoped by usng the method o the reducton o order or the contro equaton and artca boundary condtons. It s proved that the nte derence scheme s stabe and convergent wth the order n space and order 3/ n tme under an energy norm. A non-homogeneous numerca exampe demonstrates the uncondtona stabty and the accuracy o the agorthm. Keywords: non-homogeneous heat equaton; unbounded doman; artca boundary condton; nte derence method; error estmate. Introducton Heat equaton arses rom many eds or exampes the heat transer ud dynamcs astrophyscs nance or other areas o apped mathematcs. For the probem on unbounded domans Han and Huang () n [] and [] derved an exact artca boundary condton obtaned the correspondng numerca souton or heat equatons by FM and FDM n one-dmensona and two-dmensona cases showed the eectveness wth numerca exampes but had no error estmate. Wu and Sun (4) n [3] apped the artca boundary condton n [] to nvestgate the numerca souton o heat equaton on the sem-nnte nterva provng the uncondtona stabty and convergence wth order n space and the order 3/ n tme under an energy norm. In ths paper we consder the oowng Cauchy probem o one-dmensona non-homogeneous heat equaton: ut σ u xx = ( x t ) < x < + < t T u= ( x ) φ ( x < x < + u ( x t ) when x + t T. a Correspondng author : zhengq@ncut.edu.com 6. The authors - Pubshed by Atants Press 387 () () (3)
where σ s duson coecent determnng heat conducton rate densty and heat capacty o the matera. The nta term φ ( x) and the source term () xt have compact supports n space. Introducng artca boundares such that { ( )} [ Γ = {( xt ) x = x t T} and Γ r = {( xt ) x = xr t T } su p p xt x xr ] [ T] and su p p { ( x )} [ x x r ]. φ Accordng to the work o Han and Huang n [] and Wu and Sun n [3] we can reduce the nta probem to an nta-boundary probem on a bounded doman: u v σ = t x ( xt) x < x < x r < t T (4) u v = x x < x < x r (5) ux ( ) = φ( x x < x < x r (6) vx ( t) = t ux ( λ) dλ < t T λ t λ vx ( r t) = t ux ( r λ) dλ < t T. λ t λ (7) (8) The nte derence scheme and correspondng error estmaton We dscretze the above probem as oows. Dvde the doman [ x x r ] [ T ] nto an M N mesh wth the spata sze h = ( xr x )/ M n x drecton and the tme step sze τ = T / N n t drecton respectvey where M and N are postve ntegers. Grd ponts ( x t n ) are dened by x = x + h or M and tn = nτ or n N. Moreover denote n be the numerca souton o uxt () at ( x t n ) and ntroduce the notatons: n n n n n n n = ( + ( ( = + δ x = h n n n n n δx = ( ( + + δt = h τ n n n n n M n = ( + ( h ( ). = + = = n Dene the grd uncton: u = ux ( tn M n. The notatons o v are smar. Appyng nte derence method or (4)-(8 we have 388
δ tu σδ xv M n v δ xu M n δ tu σδ xv M n v δ xu M n u = φ( x (9) () () () (3) 3 3 where r ( τ + h r h s τ s τ M c are constants ndependent o τ and h. Omttng the sma error terms n (8)-(3 we construct a nte derence scheme: n and c and δ t σδ xv = M n V δ x = M n k V [ ( ) = a a a ] n k k k = k V [ ( ) M = a M a a ] n k k k M = (4) (5) (6) (7) φ( x M. = (8) Theorem. The nte derence scheme (4)-(8) s equvaent to the oowng scheme (9)-(3) wthout ntermedate varabe: ( δt + δt ) σδ x = + + + (9) δ σ k t + a ak ak δx = h k = [ ( ( ) ) ] () δ σ k t + a M an k an km x M + δ = M h M k = [ ( ( ) ) ] () 389
= φ( x () where am = ( m+ m m = 3 L. (3) τ We can obtan the stabty and the convergence o the nte derence scheme usng Gronwa nequaty as oows by usng the methods n [4] and [5]. Theorem. Let max ( xt ) 3 c n 3be a postve constant.{ } x x xr t T be the Souton o the derence scheme (4)-(8 uxt () be the souton o probem (4)-(8). Then n τ exp( )( c 3 TL n T / + = L τ τ τ 3 n n T c T σt u exp( ) [ + ( σ + ) L]( τ + h ) n= L [T/ τ] τ τ π where c = max{ c c c3}. 3 Numerca exampe We ntroduce the oowng dentons: L -error: n n h τ = max u M L -order: hτ og ( h τ / h τ L -error: M n n h τ = h [ u ] = L -order: hτ og ( h τ / h τ xampe.consder the oowng nonhomogeneous heat equaton: ( ) xt ut σuxx = σ 4 x e < x < + < t T x ux ( ) = e < x< + uxt ( ) when x + t T. x t whch has the exact souton uxt () = e.the support o φ( x) and () xt s compact on the computatona doman [ x x ] = [ 5 5]. r Tabe. Convergence w.r.t. τ o xampe or σ = and h = τ 3 4. T= T= M N L -error order L -error order N L -error order L -error order 56 4.7676e-3 ----- 4.6564e-3 -----.55e-3 -----.77e-3 ----- 95.699e-3.5574.69e-3.533 4 7.536e-4.5836 7.586e-3.5579 59 4 5.7385e-4.497 5.763e-4.493 8.694e-4.5.58e-4.54 67 8.98e-4.565.e-4.56 6 9.78e-5.54 8.86e-5.5 39
Tabe. Convergence w.r.t. h o xampe or σ = and τ = h 4 3. T= T= M N L -error order L -error order N L -error order L -error order 5 7.996e- -----.48e- ----- 4 9.97e-3 ----- 9.743e-3 ----- 5 7 5.6867e-3.875 5.787e-3.9734 34.54e-3.8393.4764e-3.87 4.49e-3.9998.436e-3.9955 84 6.348e-4.93 6.983e-4.979 7 3.544e-4.43 3.578e-4.3 4.565e-4.5.546e-4.3 We can see n Tabe that the order o convergence agrees wth the theoretca order 3/ n tme 3 3 durng the derent tme perods snce Theorem gves O( τ + h ) = O( τ ) or h = τ 3 4. Tabe ustrates the theoretca order o convergence snce Theorem gves O( τ + h) = Oh ( ) or τ = h 4 3. In addton the resuts when T= and T= show the uncondtona stabty o the proposed agorthm. 3.5 x -3 3 3 Reatve L-error.5.5 r=.5 r= r=.5...3.4.5.6.7.8.9 t Fgure. Reatve L -error o xampe durng t under derent mesh ratos or σ = and M =. We can aso see n Fgure that the numerca resut s uncondtonay stabe durng the computatona tme perod regardess o the mesh rato στ r =. h 4 Concusons The numerca souton to the non-homogenous heat equaton n the unbounded doman s consdered. Smar to the numerca methods o non-homogenous heat equaton n the haunbounded doman n [] and the nvoved homogenous heat equaton n unbounded doman n [5] we convert the orgna probem to an nta-boundary vaue probem by usng two exact artca boundary condtons. Then the nte derence scheme s obtaned by the method o reducton o order. The stabty and the correspondng error estmate o (9)-(3) are gven n Theorem. A non- 39
homogenous exampe demonstrates the uncondtona stabty and the accuracy o the proposed method. Acknowedgement Ths work s supported by Natona Natura Scence Foundaton o Chna. (479). Reerences. H.-D. Han Z.-Y. Huang. A cass o artca boundary condtons or heat equaton n unbounded domans Comput. Math. App. 43 () 889-9.. H.-D. Han Z.-Y. Huang xact and approxmatng boundary condtons or the paraboc probemson unbounded domans Comput. Math. App. 44 () 655-666. 3. X.-N. Wu Z.-Z. Sun Convergence o derence scheme or heat equaton n unbounded domansusng artca boundary condtons App. Numer. Math. 5 (4) 6-77. 4. H.-D. Han X.-N. Wu Artca Boundary Method Bejng: Tsnghua nversty Press/SprngerPress. 5. Q. Zheng L. Fan X.-Z. L An artca boundary method or Burgers' equaton n the unboundeddoman CMS: Comput. Mode. ng. Sc. (4) 445-46. 39