140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton ssues of dscretzaton methods for advecton-dffusonreacton problems are gven n the monograph by Hundsdorfer and Verwer Hundsdorfer, Verwer 003. Very nterestng applcatons of these results for bomedcal problems are descrbed by Gersch and Chaplan Gersch, Chaplan 006, Gersch and Verwer Gersch, Verwer 00 see also references gven n these publcatons. Here we propose to use a lnearzed mplct backward Euler method for the approxmaton of the dffuson-reacton subproblems and the explct forward Euler method for soluton of the advecton subproblem. We have restrcted to the frst order methods due to ther robust stablty. Note that our man goal s to nvestgate the nfluence of a possble ll-posedness of the PDE system to the asymptotcal behavour of the soluton. Appendx B.1. The method of lnes. Dscretzaton n space We use the method of lnes MOL approach see, Gersch, Chaplan 006; Hundsdorfer, Verwer 003. At the frst step we approxmate the spatal dervatves n the PDE by applyng robust and accurate approxmatons targeted for specal physcal processes descrbed by dfferental equatons. Doman [0, 1] s covered by a dscrete unform grd ω h = {x : x = h, = 0,..., N 1}, x N = 1 wth the grd ponts x. On the semdscrete doman ω h k [0, T ] we defne functons U t = Ux, t, V t = V x, t, = 0,..., N 1, here U, V approxmate solutons ux, t, vx, t on the dscrete grd ω h at tme moment t. We also defne the forward and backward space fnte dfferences wth respect to x: x U = U +1 U h, x U = U U 1. h Usng the fnte volume approach, we approxmate the dffuson and reac-
APPENDIXES 141 ton terms n the PDE system 3.1 by the followng fnte dfference equatons: A DR1 U, = D x x U + γru 1 U, U p A DR V, U, = x x V + γ 1 + βu p V. B.57 The stencl of the dscrete scheme requres to use functons defned outsde of the grd ω h. We apply perodcty boundary condtons 3.3 to defne dscrete functons for < 0 or N: U = U N, U N 1+ = U 1 for > 0. B.58 The advecton term n equaton 3.1 depends on the varable velocty ax, t := χ 1 + αv v x, therefore the maxmum prncple s not vald for the respectve transport equaton. But problem 3.1 stll has non-negatve solutons, and ths property can be preserved on the dscrete level by applyng proper upwndng approxmatons. The dscrete spatal approxmaton of the velocty s computed by a + 1 t = χ 1 + αv + V +1 / xv. In the followng we consder the upwnd-based dscrete fluxes Gersch, Chaplan 006; Hundsdorfer, Verwer 003: [ F T U, a, + 1/ = a + 1 U + ψθ ] U +1 U, a+ 1 0, B.59 F T U, a, + 1/ = a + 1 wth the Koren lmter functon ψθ = max [ U+1 + ψ1/θ +1 ] U U +1, a+ 1 0, mn 1, 1 3 + 1 6 θ, θ. The lmter ψ depends on the smoothness montor functon θ = U U 1 U +1 U. < 0,
14 APPENDIXES For ψ = 0 we get the standard frst-order upwnd flux F T UW U, a, + 1/ = max a + 1, 0 U + mn a + 1, 0 U +1. Let us denote the dscrete advecton operator as A T U, V, = 1 h FT U, a, + 1/ F T U, a, 1/. Then we obtan a nonlnear ODE system for the evaluaton of the approxmate sem-dscrete solutons du = A T U, V, + A DR1 U,, x ω h, B.60 dv = A DRV, U,. Appendx B.. Operator splttng methods In order to develop effcent solvers n tme for the obtaned large ODE systems we apply the splttng technques. They take nto account the dfferent nature of the dscrete operators defnng the advecton A T U, V, and the dffuson-reacton A DR1 U,, A DR V, U, terms. The system resolvng the sem-dscrete advecton process can be solved very effcently by usng explct solvers, whle the dffuson-reacton sem-dscrete system s stff and t requres an mplct treatment. Also we are nterested n preservng at the dscrete level the postvty and/or boundedness of the soluton, f such propertes hold for the dfferental ODE system. Frst we consder the symmetrcal splttng method also known as the Strang splttng Strang 1968. Gven approxmatons U n, V n at tme t n, we
APPENDIXES 143 compute solutons at t n+1 = t n + by the followng scheme: du = A T U, V n,, U t n = U n, t n t t n+ 1 = t n + /, B.61 du du dv = A DRV, U,, = A DR1 U,, U t n = U n+ 1, t n t t n+1, B.6 = A T U, V n,, U t n = U n+1, t n+ 1 t t n+1 V t n = V n, t n t t n+1. B.63 Here we also have spltted the gven ODE system nto two blocks wth respect to U and V functons. Lemma 1. Solutons of the splttng ODE problem B.61 B.63 are nonnegatve f U n 0 and V n 0 for all x ω h. Proof. The proof for the advecton subsystems follows from the constructon of the dscrete fluxes by usng the upwndng technque. The proof for the dffuson-reacton subsystems follows from the lemma n Gersch, Verwer 00 that the soluton of an ntal value problem for systems of ODEs dy = F t, Y t, t 0, Y 0 = Y 0 s non-negatve f and only f for all t and any vector V R m and all 1 m v = 0, v 0 for all = f t, V 0. It s easy to see that for the dffuson-reacton subsystems the dffuson parts of the matrces are dagonally domnant and all off-dagonal entres are nonpostve. For the reacton parts the requred estmates are also trvally satsfed. Lemma. If 0 U n C for all x ω h, then a soluton of the splttng ODE problem B.6 s also bounded U maxc, 1. Proof. Here we use the fact that U = 1 s a stable attractor of the reacton functon. Let U n = max U n. Then t follows from the defnton of A DR1 U, that n the worst case du > 0, f C < 1, du = 0, f C = 1,
144 APPENDIXES and The lemma s proved. du < 0, f C > 1. Appendx B.3. Numercal ntegraton of ODEs There are many numercal ntegraton methods for soluton of non-stff and stff ODEs. For detaled dscussons of these schemes we refer the reader to Gersch, Chaplan 006; Harer, Norset, Wanner 1993; Harer, Wanner 1996; Hundsdorfer, Verwer 003. Let ω be a unform tme grd ω = {t n : t n = n, n = 0,..., M, M = T f }, here s the tme step. For smplcty ths step sze s taken constant. In the followng, we consder numercal approxmatons U n, V n to the exact soluton values ux, t n, vx, t n at the grd ponts x, t n ω h ω. Remark 1. In Baronas, Šmkus 011, the explct forward Euler scheme s used to solve problem 3.1. Snce no detals are gven n Baronas, Šmkus 011 on approxmatons of spatal dervatves, we use dscrete operators ntroduced n prevous sectons and wrte the explct forward Euler scheme as: U n+1 V n+1 U n V n = A T U n, V n, + A DR1 U n,, = A DR V n, U n,. We note that ths scheme can be wrtten as a splttng algorthm: U n+ 1 U n U n+1 U n+ 1 V n+1 V n = A T U n, V n,, = A DR1 U n,, = A DR V n, U n,.
APPENDIXES 145 Thus the explct Euler scheme can be consdered as a specal case of splttng algorthms. Despte easy mplementaton and good parallel propertes of explct algorthms, the man drawback of the explct Euler method s that due to the condtonal stablty we must restrct the ntegraton step to Ch for stff dscrete dffuson-reacton subproblems. The Rosenbrock and mplct Runge-Kutta methods are successfully appled for ntegraton of a stff part of the splttng semdscrete-scheme,.e. dffuson reacton equatons B.6, B.63, see Gersch, Chaplan 006; Gersch, Verwer 00; Hundsdorfer, Verwer 003. Here we propose to use a lnearzed mplct backward Euler method for the approxmaton of the dffuson-reacton subproblems and the explct forward Euler method for soluton of the advecton subproblem. We have restrcted to the frst order methods due to ther robust stablty. Note that our man goal s to nvestgate the nfluence of a possble ll-posedness of the PDE system to the asymptotcal behavour of the soluton. We dscretze the semdscrete problem B.61 B.63 wth the fully dscrete scheme U n+ 1 3 U n 0.5 U n+ 3 U n+ 1 3 U n+1 U n+ 3 0.5 V n+1 V n = A T U n, V n,, B.64 = D x x U n+ 3 + γru n+ 1 3 1 U n+ 3, B.65 = A T U n+ 3, V n,, B.66 = A DR V n+1, U n+1,. B.67 We apply two splttngs of the advecton term, because then we use only half of the splttng step sze for the explct method. Ths doubles the stablty and postvty domans of the explct method see Gersch, Verwer 00. Next we prove that statements of Lemma 1 and hold also for solutons of the fully dscrete fnte dfference scheme B.64 B.67 Lemma 3. For a suffcently small tme step 0 solutons of the fnte dfference scheme B.64 B.67 are non-negatve f U n 0 and V n 0 for all x ω h. Proof. The proof for the advecton problems B.64 and B.66 follows from
146 APPENDIXES the constructon of the dscrete fluxes by usng the upwndng technque and selecton of a suffcently small tme step 0. The proof for the dffuson-reacton problems B.65 and B.67 follows from the maxmum prncple Samarsk 001. For example, consder problem B.65. We assume, that U n+ 3 = mn U n+ 3. 0 <N We wrte the dscrete equaton B.65 for U n+ 3 1 + γru n+ 1 3 n an explct form U n+ 3 = 1 + γr U n+ 1 3 + D h U n+ 3 +1 + U n+ 3 1 U n+ 3 Snce U n+ 1 3 0 and U n+ 3 ±1 U n+ 3 we get that U n+ 3 0. Lemma 4. If 0 U n+ 1 3 C for all x ω h, then a soluton of the fnte dfference scheme B.65 s also bounded Proof. U n+ 3 maxc, 1, x ω h. The proof s based on the maxmum prncple and a specal form of the dscrete reacton term. Frst, we consder the case C 1. Let U n+ 3 = max U n+ 3. Then t follows from B.65 that 1+γrU n+ 1 3 U n+ 3 1+γr U n+ 1 3 = U n+ 3 Next we consder the case C > 1. Then we get that 1 n+ 3 1 + γru 1 + γru n+ 1 3 1. U n+ 3 The lemma s proved. 1 n+ 3 1 + γru 1 + γru n+ 1 3 = 1 + U n+ 1 3 1 1 + γru n+ 1 3 < C.