COMPACT IMPLICIT INTEGRATION FACTOR METHODS FOR A FAMILY OF SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS. Lili Ju. Xinfeng Liu.

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1 DISCRETE AND CONTINUOUS doi:13934/dcdsb DYNAMICAL SYSTEMS SERIES B Volume 19, Number 6, August 214 pp COMPACT IMPLICIT INTEGRATION FACTOR METHODS FOR A FAMILY OF SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS Lili Ju Department of Mathematics, Universit of South Carolina Columbia, SC 2928, USA and Beijing Computational Science Research Center Beijing, 184, China Xinfeng Liu Department of Mathematics, Universit of South Carolina Columbia, SC 2928, USA Wei Leng State Ke Laborator of Scientific and Engineering Computing Chinese Academ of Sciences, Beijing, 119, China (Communicated b Qing Nie) Abstract When developing efficient numerical methods for solving parabolic tpes of equations, severe temporal stabilit constraints on the time step are often required due to the high-order spatial derivatives and/or stiff reactions The implicit integration factor (IIF) method, which treats spatial derivative terms eplicitl and reaction terms implicitl, can provide ecellent stabilit properties in time with nice accurac One major challenge for the IIF is the storage and calculation of the dense eponentials of the sparse discretization matrices resulted from the linear differential operators The compact representation of the IIF (ciif) can overcome this shortcoming and greatl save computational cost and storage On the other hand, the ciif is often hard to be directl applied to deal with problems involving cross derivatives In this paper, b treating the discretization matrices in diagonalized forms, we develop an efficient ciif method for solving a famil of semilinear fourth-order parabolic equations, in which the bi-laplace operator is eplicitl handled and the computational cost and storage remain the same as to the classic ciif for 21 Mathematics Subject Classification 65M6, 35K25 Ke words and phrases Integration factor method, compact representation, implicit scheme, bi-laplace operator, semilinear fourth-order parabolic equations L Ju s research is partiall supported b US National Science Foundation under grant number DMS and US Department of Energ, Office of Science, Advanced Scientific Computing Research and Biological and Environmental Research programs through the SciDAC project PIS- CEES under grant number DE-SC887-ER65393 X Liu s research is partiall supported b US National Science Foundation under grant numbers DMS and DMS138948, and Universit of South Carolina ASPIRE grant W Leng s research is partiall supported b National 863 Project of China under grant number 212AA1A39, and National Center for Mathematics and Interdisciplinar Sciences of Chinese Academ of Sciences 1667

2 1668 LILI JU, XINFENG LIU AND WEI LENG second-order problems In particular, the proposed method can deal with not onl stiff nonlinear reaction terms but also various tpes of homogeneous or inhomogeneous boundar conditions Numerical eperiments are finall presented to demonstrate effectiveness and accurac of the proposed method 1 Introduction Let Ω be an open rectangular domain in R d and a final time T > In this paper, we consider a famil of semilinear fourth-order parabolic equations taking the following form: { u = D 2 u + f(u), Ω, t [, T ], t (1) u t= = u, Ω, where D > is the diffusion coefficient Different boundar conditions such as the Dirichlet boundar condition, periodic boundar condition or Neumann boundar condition will all be studied in this paper One of major difficulties in numericall solving such equations is how to efficientl handle the bi-laplace operator coupled with the stiff nonlinear reaction term f(u) In general, the time step size relies heavil on stiffness of the reactions and treatment of the high-order derivatives Integration factor (IF) or eponential differencing time (ETD) methods are among the effective approaches to deal with temporal stabilit constraints associated with reaction-diffusion equations [5, 14, 15] B treating the highest order spatial derivatives eactl in time direction, the IF or ETD methods are able to achieve ecellent temporal stabilit [6, 7, 14, 16] To deal with additional stabilit constraints from the stiff reactions, an implicit integration factor (IIF) method [21] was developed for implicit treatment of the stiff reactions In the IIF approach, the diffusion term is solved eactl like the IF method while the nonlinear equations resulted from the implicit treatment of reactions are decoupled from the diffusion term to avoid solving large nonlinear sstems involving both diffusions and reactions, such as in the standard implicit method for reaction-diffusion equations The IIF method has ver nice stabilit, for eample, its second-order scheme being linearl unconditionall stable [21] In the IF based methods the dominant computational cost arises from storage and calculation of eponentials of matrices resulting from discretization of the linear differential operators To overcome this difficult in high spatial dimensions, compact representation of the discretization matrices was introduced in the contet of the IIF method [2] In the compact implicit integration factor (ciif) method in two dimensions, the discretized solutions are represented in a matri form rather than a vector while the discretized diffusion operator are represented in matrices of much smaller size than the ones for the IIF while still preserving the stabilit propert of the IIF In three or higher dimensions, the ciif is significantl more efficient in both storage and computational cost since the discretized matri for each spatial direction has the same size as the classic IIF in one dimension In addition, the ciif method is robust in its implementation and integration with other spatial and temporal discretization algorithms For eample, it can handle general curvilinear coordinates as well as combining with adaptive mesh refinements in a straightforward fashion [18] One also can appl the ciif to stiff reactions and diffusions while using other specialized hperbolic solvers (eg WENO methods [19, 12]) for convection terms to solve reaction-diffusion-convection equations efficientl [26] On the other hand, the ciif is often hard to be directl applied to deal with problems involving cross derivatives In this paper, we develop a ciif method for

3 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1669 solving a famil of semilinear fourth-order parabolic equations, in which the bi- Laplace operator is eplicitl handled In this approach we use standard central finite differences for spatial discretization coupled with compact implicit integration factor methods for time discretization The discretized matrices arising from a compact representation of the diffusion operator need to be diagonalized just once and pre-calculated before each time step iteration We also discuss several other ke issues in the construction and implementation of the ciif method such as fast evaluations of matri-vector multiplications and stable and accurate incorporations of inhomogeneous boundar conditions, while how to deal with inhomogeneous boundar conditions with ciif was not addressed at all in previous studies [2, 21] This new approach is found to have similar stabilit properties and computational cost as the ciif for second-order problems In the remainder of the paper, we first derive and analze the ciif method for the semilinear fourth-order parabolic equation (1) with various homogeneous and inhomogeneous boundar conditions (Dirichlet, Periodic and Neumann boundar conditions) in two dimensions in Section 2, then present its etension to three dimensions in Section 3 Numerical simulations given in Section 4 ehibit ecellent performance of the proposed ciif method Finall a conclusion is drawn in Section 5 2 Compact implicit integration factor methods in two dimensions In this section, we derive compact implicit integration factor (ciif) methods for the model equation (1) in two dimensions Suppose Ω = { b < < e, b < < e } The equation (1) can be written as u t = D(u + 2u + u ) + f(u), (, ) Ω, t [, T ] (2) 21 The problem with Dirichlet boundar conditions We first consider the problem with a Dirichlet boundar condition { u = g 1, (, ) Ω, t [, T ], (3) u = g 2, (, ) Ω, t [, T ] Let us discretize the spatial domain b a rectangular mesh which is uniform in each direction as follows: ( i, j ) = ( b + ih, b + jh ) where h = ( e b )/N, h = ( e b )/N and i N and j N We will use the central finite difference discretization scheme with second-order accurac for the three spatial derivative terms: u, u and u Set u i,j = u(t, i, j ) for i N and j N Define δ 2 u i,j = u i+1,j 2u i,j + u i 1,j h 2, δu 2 i,j = u i,j+1 2u i,j + u i,j 1 h 2 Then we can write the semi-discretization of (2) b the second-order central difference scheme in the following form: du i,j dt for < i < N and < j < N = D(δ 2 δ 2 u i,j + 2δ 2 δ 2 u i,j + δ 2 δ 2 u i,j ) + F(u i,j ), (4)

4 167 LILI JU, XINFENG LIU AND WEI LENG Denote the set of unknowns as u 1,1 u 1,2 u 1,N 1 u 2,1 u 2,2 u 2,N 1 U = u N 1,1 u N 1,2 u N 1,N 1 and set D U 1 = h 2 (N 1) (N 1) g 1(t,, 1) g 1(t,, 2) g 1(t,, N 1) g 1(t, N, 1) g 1(t, N, 2) g 1(t, N, N 1) g 1 (t, 1, ) g 1 (t, 1, N ) D g 1 (t, 2, ) g 1 (t, 2, N ) U 1 = h 2 g 1 (t, N 1, ) g 1 (t, N 1, N ) g 2(t,, 1) g 2(t,, 2) g 2(t,, N 1) U 2 = D h 2 g 2(t, N, 1) g 2(t, N, 2) g 2(t, N, N 1) g 2 (t, 1, ) g 2 (t, 1, N ) U 2 = D g 2 (t, 2, ) g 2 (t, 2, N ) h 2 g 2 (t, N 1, ) g 2 (t, N 1, N ) Let us further define G P P = and set A = D G (N 1) (N 1), B = D h 2 operators and as follows: (A U) i,j = (B U) i,j = h 2 G (N 1) (N 1) (A) i,l u l,j, N 1 l=1 N 1 (B) j,l u i,l l=1 Note here that these two operators are commutative, ie, B A U = A B U P P,, (N 1) (N 1) (N 1) (N 1) (N 1) (N 1) (N 1) (N 1) Define the special,,, (5)

5 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1671 After incorporating with boundar conditions, the compact representation of (4) takes the following form: du dt which can be simplified to = A (A U + U 1 + B U + U 1 ) U 2 B (A U + U 1 + B U + U 1 ) U 2 + F(U), (6) du dt = A2 U 2B A U B 2 U + E + F(U), (7) where F(U) = (f(u i,j )) (N 1) (N 1) and E = A (U 1 + U 1 ) U 2 B (U 1 + U 1 ) U 2 (8) Here A and B are diagonalizable and let us assume that A = P D P 1, B = P D P 1, (9) where D and D are diagonal matrices with the eigenvalues of A and B as the diagonal elements, respectivel Let V = P 1 P 1 U, then (7) can be transformed into dv dt = D 2 V 2 D D V D 2 V + P 1 P 1 (E + F(U)), (1) and thus dv dt + D 2 V + 2 D D V + D 2 V = P 1 P 1 (E + F(U)) (11) Suppose that the diagonal matrices are given b D = diag[d 1, d 2,, d N 1 ] and D = diag[d 1, d 2,, d N 1 ] Set H = ( h i,j ) (N 1) (N 1) with h i,j = 2d i d j Here we define an operation (e ) b taking element-b-element eponentials as the following, (e ) H = (e h i,j ) (N 1) (N 1) Define another operator for element-b-element multiplication between two matrices with the same size b the following, (M L) i,j = (m i,j l i,j ), where M = (m i,j ), L = (l i,j ) Then direct etension on (11) leads to [ ( )] e D 2 t e D 2 dv t dt + D 2 V + 2 D D V + D 2 V (e ) Ht ) = [ ] e D 2 t e D 2 t P 1 P 1 (E + F(U)) (e ) Ht ), (12) which can be simplified b d(v (e ) Ht ) dt = P 1 (E + F(U))) (e ) Ht, (13) where H = (h i,j ) (N 1) (N 1) with h i,j = (d i + d j )2 Let us discretize the time as t n = n t, n =, 1,, N t with t = T/N t Then taking integration on both sides of (13) from t n to t n+1 gives V n+1 (e ) H t V n = P 1 (E + F(U))) (e ) Hτ dτ (14)

6 1672 LILI JU, XINFENG LIU AND WEI LENG Thus we have ( V n+1 = V n + P 1 ) (E + F(U))) (e ) Hτ dτ (e ) H t (15) Finall b substituting V b U in (15), the numerical scheme can be achieved for solving the model equation (1) from t n to t n+1, ( U n+1 = P P U n) (e ) H t + P 1 P 1 (E + F(U))) (e ) H( t τ) dτ ) (16) Net we use multistep schemes to accuratel evaluate the integral which appears on the right-hand-side of (16), ie, P 1 (E+F(U))) (e ) H( t τ) dτ 211 Multistep approimations of the integral from the boundar conditions To evaluate the integral resulted from the inhomogeneous boundar terms P 1 E(t n + τ)) (e ) H( t τ) dτ, we need to be careful since (e ) H( t τ) contains entries which deca with highl different speeds along the time, and E(t n + τ) involves the factors of 1/h 4 and 1/h 4 which could quickl amplif errors arising from the time discretization, which would cause severe numerical instabilit To overcome this difficult, here we will appl an elegant approach proposed in [13], which will be described with details in the following Let and ie, W(t n + τ) = (w i,j (t n + τ)) (N 1) (N 1) = P 1 Q W = (q W i,j) (N 1) (N 1) = q W i,j = P 1 E(t n + τ), W(t n + τ) (e ) H( t τ) dτ, e hi,j( t τ) w i,j (t n + τ) dτ It is reasonable to assume that W(t) changes smoothl along the time Thus based on the values of W(t) at t n+1, t n,, t n+1 r1, we use the Lagrange interpolation polnomial P W r 1 (τ) of degree r 1 to approimate W(t n + τ) (the all can be calculated from the given inhomogeneous boundar conditions): with ω r1,s(τ) = r 1 1 l= 1 l s P W r 1 (t n + τ) = Thus we can approimate q W i,j b q W i,j r 1 1 s= 1 w i,j (t n s ) r 1 1 s= 1 τ + l t Then we have (l s) t ω r1,s(τ)w(t n s ), (17) W(t n + τ) P W r 1 (τ) + O( t r1+1 ) (18) e hi,j( t τ) ω r1,s(τ) dτ = r 1 1 s= 1 w i,j (t n s )α (r1,s) i,j, (19)

7 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1673 where α (r1,s) i,j = e hi,j( t τ) ω r1,s(τ) dτ We remark that α (r1,s) i,j is independent of the time steps since a uniform time step size t is used here The main idea here is to evaluate α (r1,s) i,j eactl to avoid an loss of accurac and numerical instabilit For instance, the values of {α (r1,s) i,j } for r 1 =, 1, 2 are listed as below r 1 = : r 1 = 1 : r 1 = 2 : α (, 1) i,j = φ ; (2) α (1, 1) i,j = φ 1, α (1,) i,j = φ φ 1 ; (21) α (2, 1) i,j = 1 2 (φ 1 + φ 2 ), α (2,) i,j = φ φ 2, α (2,1) i,j = 1 2 (φ 1 φ 2 ); (22) where φ = 1 (1 e hi,j t ), φ 1 = 1 (1 φ h i,j h ij t ), φ 2 = 1 (1 2φ 1 h ij t ), h i,j, φ = t, φ 1 = t 2, φ 2 = t 3, h i,j = (23) Define S r1,s = (α (r1,s) i,j ) (N 1) (N 1), then we can have the following approimation for the integral arising from the inhomogeneous boundar conditions P 1 E(t n + τ)) (e ) H( t τ) dτ Such approimation is (r 1 + 1)-th order accurate in time r 1 1 s= 1 W n s S r1,s (24) 212 Implicit multistep approimations of the integral from the nonlinear reaction term For evaluation of the integral resulted from the nonlinear term P 1 F(U(t n + τ))) (e ) H( t τ) dτ, we ma still use a similar multistep approach to obtain an implicit approimation for the nonlinear term for the purpose of stabilit However, since F(U(t n + τ)) itself does not involve an term related to spatial mesh size and e hi,j( t τ) 1 for τ < t, thus we can directl appl the Lagrange interpolation polnomial of degree r 2 at t n+1, t n,, t n+1 r2 τ))) (e ) H( t τ) as proposed in [2] It is not difficult to find that P 1 t to the whole integrand F(U(t n + τ))) (e ) H( t τ) dτ r 2 1 s= 1 β r2,s r 2 1 = t β r2,s s= + tβ r2, 1 P 1 P 1 P 1 P 1 F(U(t n + F(U n s)) (e ) (s+1)h t F(U n s)) (e ) (s+1)h t P 1 F(U n+1), (25)

8 1674 LILI JU, XINFENG LIU AND WEI LENG with β r2,s = values of {β (r2,s) i,j ω r2,s(τ) dτ, which is (r 2 + 1)-th order accurate For eample, the } for r 2 =, 1, 2 are presented as below r 2 = : β (, 1) = 1; (26) r 2 = 1 : β (1, 1) = 1 2, β(1,) i,j = 1 2 ; (27) r 2 = 2 : β (2, 1) = 5 12, β(2,) i,j = 2 3, β(2,1) i,j = 1 12 (28) Note {β (r2,s) i,j } are constant across all points 22 The compact implicit integration factor schemes Now put (24) and (25) back into (16), and we obtain a compact implicit integration factor (ciif) scheme that is second-order accurate in space and at least (min{r 1, r 2 } + 1)-th order accurate in time: ( U n+1 = P P P 1 U n) (e ) H t + r 2 1 S r1,s + t β r2,s s= P 1 r 1 1 s= 1 P 1 F(U n s)) (e ) (s+1)h t) E n s) + tβ r2, 1 F(U n+1 ) (29) Clearl the ciif scheme (29) is completel eplicit when there isn t the nonlinear reaction term in the model problem (1), ie, f(u) It is speciall worth noting that the nonlinear solution process (that is needed when f(u) is nonlinear) in the ciif scheme (29) is point-wise and thus ver efficient b using some Newton-tpe iterative method for scalar functions If we appl the same approimation approach for the first integral as discussed in Sections 211 to the second integral, then we obtain a much more epensive implicit scheme since the nonlinear solution process becomes global When there is a function g(, ) included in the right hand side of the model equation (1), it can be treated eactl the same wa as that for inhomogeneous boundar conditions We present the detailed scheme with some tpical choice of r 1 and r 2 in the following r 1 = r 2 = (first-order in time): ( U n+1 = P P + r 1 = r 2 = 1 (second-order in time): ( U n+1 = P P + P 1 P 1 U n) (e ) H t P 1 E n+1) S, 1 ) + tf(u n+1 ); (3) P 1 (U n + t 2 F(U n))) (e ) H t E n+1) S 1, 1 + P 1 E n) S 1, ) + t 2 F(U n+1); (31)

9 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1675 r 1 = r 2 = 2 (third-order in time): ( U n+1 = P P P 1 S 2, 1 + P 1 P 1 (U n + 2 t 3 F(U n))) (e ) H t t 12 F(U n 1)) (e ) 2H t + P 1 E n+1) ) E n 1) S 2, + P 1 E n 1) S 2,1 + 5 t 12 F(U n+1) (32) Remark 1 As discussed in [13, 24], the computational compleities of P V, P 1 V, P V, P 1 V for an (N 1) (N 1) arra V can be reduced from O(N 3 ) to O(N 2 log 2 (N)) with N = ma{n, N } through the use of fast Discrete Sine Transform (DST) if we choose ), k = 1, 2,, N 1, d k = 4D h 2 d k = 4D h 2 ( kπ sin 2 2N sin 2 ( kπ 2N ), k = 1, 2,, N 1 Thus the overall cost of the proposed ciif scheme in two dimensions is O(N 2 log 2 N) per time step In addition, the overall computational storages of the proposed ciif scheme are in the order of O(N 2 ), O(N 2 ) or O(N N ) 23 Other tpe of boundar conditions 231 Periodic boundar conditions If a periodic boundar condition such as u(t, b, ) = u(t, e, ), u (t, b, ) = u (t, e, ), [ b, e ], u(t,, b ) = u(t,, e ), u (t,, b ) = u (t,, e ), [ b, e ], (33) u(t, b, ) = u(t, e, ), ( u) (t, b, ) = ( u) (t, e, ), [ b, e ], u(t,, b ) = u(t,, e ), ( u) (t,, b ) = ( u) (t,, e ), [ b, e ], is imposed on the model equation (1) and here t [, T ] For this case, the unknown solutions can be denoted b u, u,1 u,n 1 u 1, u 1,1 u 1,N 1 U = (u i 1,j 1 ) N N =, and set G P P = u N 1, u N 1, u N 1,N P P, N N h 2 h 2 and A = D G N N, B = D G N N The ciif scheme for the periodic boundar condition takes the same from as (29) ecept that we have E = here

10 1676 LILI JU, XINFENG LIU AND WEI LENG Remark 2 In the case of periodic boundar conditions, the computational compleities of P V, P 1 V, P V, P 1 V can be reduced from O(N 3 ) to O(N 2 log 2 (N)) through the use of fast Discrete Fourier Transform (DFT) [13, 25] if we choose d k = 4D ( ) (k 1)π h 2 sin 2, k = 1, 2,, N, N ), k = 1, 2,, N ( d (k 1)π k = 4D h 2 sin 2 N 232 Neumann boundar conditions If a Neumann boundar condition such as u = u b 1, = b 1, (, ) Ω, t [, T ], u = b 2, u = b 2, (, ) Ω, t [, T ], (34) is imposed on the model equation (1), then we ma denote the unknown solutions b u, u,1 u,n u 1, u 1,1 u 1,N U = (u i 1,j 1 ) (N+1) (N +1) =, and set and A = D h 2 but with U 1 = 2 D h U 1 = 2 D h U 2 = 2D h G P P = u N, u N, u N,N P P G (N+1) (N +1), B = D h 2 G (N+1) (N +1) We again have (N +1) (N +1) E = A (U 1 + U 1 ) U 2 B (U 1 + U 1 ) U 2, (35) b 1(t,, ) b 1(t,, 1) b 1(t,, N ), b 1(t, N, ) b 1(t, N, 1) b 1(t, N, N ) (N +1) (N +1) b 1 (t,, ) b 1 (t,, N ) b 1 (t, 1, ) b 1 (t, 1, N ), b 1 (t, N, ) b 1 (t, N, N ) (N +1) (N +1) b 2(t,, ) b 2(t,, 1) b 2(t,, N ), b 2(t, N, ) b 2(t, N, 1) b 2(t, N, N ) (N +1) (N +1)

11 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1677 U 2 = 2D h b 2 (t,, ) b 2 (t,, N ) b 2 (t, 1, ) b 2 (t, 1, N ) b 2 (t, N, ) b 2 (t, N, N ) (N +1) (N +1) The ciif scheme for the Neumann boundar condition again takes the same form as (29) Remark 3 In the case of Neumann boundar conditions, the computational cost of P V, P 1 V, P V, P 1 V again can reduced from O(N 3 ) to O(N 2 log 2 (N)) through the reflective etension and the use of fast Discrete Fourier Transform (DFT) [13, 25] 24 Linear stabilit analsis We test the linear stabilit of the proposed ciif schemes with the following linear equation u t = (q 1 + q 2 ) 2 u + du, where d >, (36) where q 1 and q 2 represent the coefficients for the bi-laplace term in, directions, respectivel As an eample, we consider the second-order scheme (31) without terms from boundar conditions In order to obtain the stabilit conditions, we first appl the second-order scheme to solve the linear equation (36), then substitute u n = e inθ into the scheme This results in e iθ = e (q1+q2)2 t ( ) λ λeiθ, where λ = d t Let us solve the real part (λ r ) and the imaginar part (λ i ) of λ, and this leads to 2(1 e λ r = 2(q1+q2)2 t ) (1 e (q1+q2)2 t ) 2 + 2(1 + cos θ)e, (q1+q2)2 t (37) λ i = 4 sin θe (q1+q2)2 t (1 e (q1+q2)2 t ) 2 + 2(1 + cos θ)e (q1+q2)2 t Since (q 1 + q 2 ) 2 >, we have the real part λ r > for all θ 2π Therefore the stabilit region lies in the whole left half of comple plane, and thus the second-order scheme (31) is A-stable For the third-order scheme as shown in (32), a similar analsis can be performed to obtain e iθ e (q1+q2)2 t λ = 5 12 eiθ e (q1+q2) t 1 (38) 2 12 e 2(q1+q2) t To investigate the stabilit region, a recurrence relation of the third order scheme is given b ( 1 5 ) 12 λ u n+1 e (q1+q2)2 t ( ) t λ u n + e 2(q1+q2)2 λ = 12 Thus the characteristic polnomial is (1 512 ) λ ω 2 e (q1+q2)2 t ( ) t λ ω + e 2(q1+q2)2 λ =, 12

12 1678 LILI JU, XINFENG LIU AND WEI LENG which has roots ω = e (q1+q2)2 t ) ( λ ± 7 12 λ2 + λ λ The stabilit region is then obtained b solving w < 1, ie, λ ± 7 12 λ2 + λ λ < 2 e(q1+q2) t (39) Obviousl the stabilit region is an increasing function of (q 1 + q 2 ) 2 t When (q 1 + q 2 ) 2 t, the stabilit region becomes the entire comple plane ecept one zero point (λ = 12/5) on the real ais For the purpose of eas illustration, the stabilit regions of the third-order scheme for (q 1 + q 2 ) 2 t =, 5, 6, 8 are plotted in Figure q t=8 q t=6 2 q t= q t= Figure 1 Stabilit regions (interior of the closed contour) for the third-order scheme (32) with q t =, 5, 6, 8, where q = (q 1 + q 2 ) 2 3 Compact implicit integration factor methods in three dimensions In this section we derive the ciif methods for the model problem (1) in three dimensions Let Ω = { b < < e, b < < e, z b < z < z e } The equation (1) can now be written as u = D(u + u + u zzzz + 2u + 2u zz + 2u zz ) t +f(u), (,, z) Ω, t [, T ] (4) In this section, we onl present the case with a Dirichlet boundar condition, and the derivation for other boundar conditions follows the same like the two dimensional case

13 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1679 Similar to solving the two dimensional sstem (2), we denote h, h, h z as the spatial mesh size, and N, N, N z as the number of grid points in,, z directions, respectivel Let A, B, and G defined the same as before in two dimensional case D We now define C = G (Nz 1) (N z 1) Set u i,j,k = u(t, i, j, z k ) and u i,j,k = h 2 z u(t, i, j, z k ) for i N, j N and k N z Denote the unknown solutions as a three-dimensional arra b U = (u i,j,k ) (N 1) (N 1) (N z 1) Define the following three-dimensional arras with a dimension (N 1) (N 1) (N z 1), D g 1 (t,, j, z k ), i = 1, U 1 = h 2 (µ i,j,k), µ i,j,k =, 1 < i < N 1, g 1 (t, N, j, z k ), i = N 1, D g 1 (t, i,, z k ), j = 1, U 1 = h 2 (µ i,j,k ), µ i,j,k =, 1 < j < N 1, g 1 (t, i, N, z k ), j = N 1, D g 1 (t, i, j, z ), k = 1, U z1 = h 2 (µ z i,j,k), µ z i,j,k =, 1 < k < N z 1, z g 1 (t, i, j, z Nz ) k = N z 1, U 2 = D g 2 (t,, j, z k ), i = 1, h 2 (νi,j,k), νi,j,k =, 1 < i < N 1, g 2 (t, N, j, z k ), i = N 1, U 2 = D g 2 (t, i,, z k ), j = 1, h 2 (ν i,j,k ), ν i,j,k =, 1 < j < N 1, g 2 (t, i, N, z k ), j = N 1, U z2 = D g 2 (t, i, j, z ), k = 1, h 2 (νi,j,k), z νi,j,k z =, 1 < k < N z 1, z g 2 (t, i, j, z Nz ) k = N z 1 With a similar analsis like two dimensional case, we can write the semi-discretization of (4) as the following compact representation, U t = A A U B B U C z C z U 2A B U 2B C z U 2C z A U + E + F(U), (41) where the operations, and z are defined as (A U) i,j,k = (B U) i,j,k = (C z U) i,j,k = (A) i,l u l,j,k, N 1 l=1 N 1 (B) j,l u i,l,k, l=1 N z 1 (C) k,l u i,j,l, l=1 (42)

14 168 LILI JU, XINFENG LIU AND WEI LENG and the three-dimensional arra E is defined correspondingl from the given inhomogeneous Dirichlet boundar conditions as E = A (U 1 + U 1 + U z1 ) U 2 B (U 1 + U 1 + U z1 ) U 2 C z (U 1 + U 1 + U z1 ) U z2 (43) Once again we observe that C is diagonalizable with C = P z Dz P 1 z where the diagonal matri D z = diag[d z 1, d z 2,, d z N ] with z 1 dz i being the eigenvalues of C Following the similar analsis in two dimensions, the compact implicit integration factor (ciif) scheme in three dimensions is given as below, ( U n+1 = P z z P P U n) (e ) H t + r 1 1 z s= 1 r t s= z P 1 β r2,s z z P 1 z P 1 z P 1 P 1 E n s) S r1,s P 1 F(U n s)) (e ) (s+1)h t) + tβ r2, 1 F(U n+1 ) (44) where H = (h i,j,k ) (with a dimension of (N 1) (N 1) (N z 1)) with h i,j,k = (d i +d j +dz k )2 and S W r 1,s = (α (r1,s) i,j,k ) (with a dimension of (N 1) (N 1) (N z 1)) are defined as a three dimensional version of (19) Remark 4 The overall cost of the proposed ciif scheme in three dimensions similarl can be reduced from O(N 4 ) to O(N 3 log 2 N) where N = ma{n, N, N z } per time step b using FFT-based fast calculations The overall storages of the proposed ciif scheme in three dimensions are in the order of O(N), 2 O(N 2 ), O(Nz 2 ) or O(N N N z ) 4 Numerical eperiments In this section we will demonstrate effectiveness and accurac of the proposed ciif scheme through numerical eamples In all simulations the differences between the numerical solutions and the eact solutions at the final time T are measured b the L 2 and L errors Eample 1 In this eample we test the ciif scheme in two dimensions using a simple linear equation, which takes the following form { u = 3 t 5π 4 2 u + 5u, (, ) Ω, t [, T ], (45) u t= = sin π( 1 4 ) cos 2π( 1 8 ), (, ) Ω, where Ω = [ 1, 1] [ 1, 1] The eact solution is given b ( u(,, t) = e 1t sin π 1 ) ( cos 2π 1 ) (46) 4 8 The Dirichlet or Neumann boundar conditions are derived accordingl from the eact solution Note that the eact solution (46) automaticall satisfies the periodic boundar conditions We set the final time T = 1 First we test the ciif scheme with r 1 = r 2 = 1 as depicted in (31) which is theoreticall second-order accurate in both space and time We run the eperiments at five different spatial and temporal resolutions Numerical results are reported Tables 1 (for Dirichlet, Neumann and Periodic boundar conditions) As epected we can clearl see the second-order convergence in both space and time for all cases

15 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1681 To test the ciif scheme with r 1 = r 2 = 2 which is third-order accurate in time, we fi a uniform fine spatial grid with N = N = 248, and conduct a series of simulations b varing time step sizes Numerical results are reported in the bottom parts of Tables 1, higher-order accurac (slightl smaller than 3) in time are clearl observed (N N ) N t L 2 Error Order L Error Order CPU (s) Dirichlet Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = 1 (16 2 ) e-1-127e-1-5 (32 2 ) e e (64 2 ) e e (128 2 ) e e (256 2 ) e e Accurac test of the ciif scheme with r 1 = r 2 = 2 (248 2 ) 8 138e-3-148e (248 2 ) e e (248 2 ) e e (248 2 ) 64 38e e (248 2 ) e e Neumann Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = 1 (16 2 ) e e+ - 4 (32 2 ) e e (64 2 ) e e (128 2 ) e e (256 2 ) e e Accurac test of the ciif scheme with r 1 = r 2 = 2 (248 2 ) 8 8e-3-115e (248 2 ) e e (248 2 ) e e (248 2 ) e e (248 2 ) e e Periodic Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = 1 (16 2 ) e-4-19e-4-2 (32 2 ) 32 17e e (64 2 ) e e (128 2 ) e e (256 2 ) e e Accurac test of the ciif scheme with r 1 = r 2 = 2 (248 2 ) 8 489e-6-489e (248 2 ) e e (248 2 ) 32 65e e (248 2 ) e e (248 2 ) e e Table 1 Errors and convergence rates at the final time T = 1 of Eample 1 (2D linear case) with inhomogeneous Dirichlet boundar condition, Neumann boundar condition and Periodic boundar condition respectivel The units of CPU times are in seconds (s)

16 1682 LILI JU, XINFENG LIU AND WEI LENG Remark 5 If eplicit methods such as the standard Runge-Kutta methods are applied to solve (45), etremel small time steps (proportional to min(h 4, h 4 )) are generall needed in order to satisf the stabilit condition [9] We have checked all the simulations in Table 1 using the second order Runge-Kutta method with the same spatial grids and time steps, none of the simulation converges due to severe stabilit requirement of Runge-Kutta methods For instance, for a tpical simulation in Table 1 with Dirichlet boundar condition with N N = and r 1 = r 2 = 1, the second order Runge-Kutta method takes CPU seconds in order to achieve the same accurac as ciif, while due to its efficienc and ecellent stabilit condition, the proposed ciif method onl takes 1113 seconds (13 14 times faster) Eample 2 Net we consider a similar linear problem in three dimensions, { u t = 2 u + 12π 4 u, (,, z) Ω, t [, T ], u t= = sin π( 1 4 ) cos 2π( 1 8 ) sin πz, (,, z) Ω, (47) where Ω = [ 1, 1] [ 1, 1] [ 1, 1] and T = 5 12π The eact solution of the above 4 equation takes the following form, ( u(,, z, t) = e 24π4t sin π 1 ) ( cos 2π 1 ) sin πz (48) 4 8 The three boundar conditions are again imposed correspondingl from the eact solution We first appl the second-order ciif scheme as presented in (44) to solve the equation (47) Similar to the two dimensional case, the second-order accurac in both time and space can be observed (see Tables 2) Due to high demanding computational cost and storage in three dimensions, we choose a uniform relativel fine spatial grid N = N = N z = 128 to test the third-order (in time) ciif scheme in time discretization Since the order accurac will be compromised between space (second-order) and time (third-order), a high order accurac (bigger than 2 but less than 3) is again observed as epected Eample 3 For this eample, we consider a bi-laplace operator with a nonlinear interaction term in two dimensions, u = 1 t 4π 2 2 u sin2 u 1, (, ) Ω, t [, T ], (49) u t= = sin π( 1 4 ) sin π( 1 8 ), (, ) Ω, where Ω = [ 1, 1] [ 1, 1] and T = 1 Inhomogeneous Dirichlet or Neumann boundar conditions are imposed along the time evolution For instance, the inhomogeneous Dirichlet boundar conditions are given b { u = e t sin π( 1 4 ) sin π( 1 8 ), u = 2π 2 e t sin π( 1 4 ) sin π( 1 8 ), and the inhomogeneous Neumann boundar conditions are given b u = πe t cos π( 1 4 ) sin π( 1 u 8 ), = πe t sin π( 1 4 ) cos π( 1 8 ), u = 2π3 e t cos π( 1 4 ) sin π( 1 8 ), u = 2π3 e t sin π( 1 4 ) cos π( 1 8 ),

17 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1683 (N N N z) N t L 2 Error Order L Error Order CPU (s) Dirichlet Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = 1 (16 3 ) e-2-488e-2-48 (32 3 ) e e (64 3 ) e e (128 3 ) e e Accurac test of the ciif scheme with r 1 = r 2 = 2 (128 3 ) 4 467e-3-437e (128 3 ) 8 839e e (128 3 ) e e (128 3 ) e e Neumann Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = 1 (16 3 ) e e (32 3 ) e e (64 3 ) e e (128 3 ) e e Accurac test of the ciif scheme with r 1 = r 2 = 2 (128 3 ) 4 51e-2-52e (128 3 ) 8 936e e (128 3 ) e e (128 3 ) 32 35e e Periodic Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = 1 (16 3 ) e-4-864e-5-2 (32 3 ) 32 25e e (64 3 ) e e (128 3 ) e e Accurac test of the ciif scheme with r 1 = r 2 = 2 (128 3 ) 4 734e-6-519e (128 3 ) 8 116e e (128 3 ) e e (128 3 ) e e Table 2 Errors and convergence rates at the final time T = 5 12π 4 of Eample 2 (3D linear case) with inhomogeneous Dirichlet boundar condition, Neumann boundar condition and Periodic boundar condition respectivel The units of CPU times are in seconds (s) for (, ) Ω, t [, T ] As an eample, the numerical solutions of the equation (49) with Dirichlet and Neumann boundar conditions at the final time T = 1 obtained b using the secondorder ciif scheme on a grid of size (N N ) N t = (256 2 ) 128 are plotted in Figure 2 For this case, we check order of accurac in time discretization b a numerical resolution stud Since the eact solution is not known for the problem (49), the approimate solution b using the third-order (in time) ciif scheme on a uniform fine grid along with a ver small time step t = 1 4, is considered as the eact solution The L 2 and L errors are measured between numerical solutions with such eact solution The results of errors and convergence rates in time discretization can be found from Table 3, which coincide with our epectation ver well

18 1684 LILI JU, XINFENG LIU AND WEI LENG Figure 2 Plot of the numerical solutions u of Eample 3 with Dirichlet boundar condition (left) and Neumann boundar condition (right) at the final time T = 1 that are obtained b using the second-order ciif scheme on a grid of size (N N ) N t = (256 2 ) 128 Eample 4 In this eample, we consider a sstem of bi-laplace equations with nonlinear interactios in two dimensions, u = 1 t 1π 2 2 u sin2 u cos v 2, (, ) Ω, t [, T ], v t = 1 5π 2 2 v cos2 v sin u 2, (, ) Ω, t [, T ], (5) u t= = sin π( 1 2 ) cos π( 1 4 ), (, ) Ω, v t= = cos π( 1 2 ) sin π( 1 4 ), (, ) Ω, where Ω = [ 1, 1] [ 1, 1] and T = 1 Inhomogeneous Dirichlet or Neumann boundar conditions are imposed along the time evolution For instance, the inhomogeneous Dirichlet boundar conditions are given b { u = e 2t sin π( 1 ) cos π( 1 ), v = 2 4 e 2t cos π( 1 ) sin π( 1 ), 2 4 u = 2π 2 e 2t sin π( 1 2 ) cos π( 1 4 ), v = 2π2 e 2t cos π( 1 2 ) sin π( 1 4 ), and the inhomogeneous Neumann boundar conditions are given b u = πe 2t cos π( 1 ) cos π( 1 u ), 2 4 = πe 2t sin π( 1 ) sin π( 1 ), 2 4 u v = πe 2t sin π( 1 ) sin π( 1 v ), 2 4 = πe 2t cos π( 1 ) cos π( 1 ), 2 4 v = 2π3 e 2t cos π( 1 ) cos π( 1 u ), 2 4 = 2π3 e 2t sin π( 1 ) sin π( 1 ), 2 4 = 2π3 e 2t sin π( 1 ) sin π( 1 v ), 2 4 = 2π3 e 2t cos π( 1 ) cos π( 1 ), 2 4 for (, ) Ω, t [, T ] For this eample, we check the order of accurac in time discretization with the second order ciif scheme b a numerical resolution stud Since the eact solution is not known for the problem (5), once again the approimate solution b using the third-order (in time) ciif scheme on a uniform fine grid along with a ver small time step t = 1 4, is considered as the eact solution Both L 2 and L errors are measured between numerical

19 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1685 (N N ) N t L 2 Error Order L Error Order Dirichlet Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = (248 2 ) e+ - 86e+ - (248 2 ) e e+ 86 (248 2 ) e e+ 11 (248 2 ) e e+ 98 Accurac test of the ciif scheme with r 1 = r 2 = 1 (248 2 ) e e+ - (248 2 ) e e+ 183 (248 2 ) e e-1 21 (248 2 ) e e Accurac test of the ciif scheme with r 1 = r 2 = 2 (248 2 ) e e+ - (248 2 ) e e (248 2 ) e e-2 31 (248 2 ) e e Neumann Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = (248 2 ) 16 26e e+ - (248 2 ) 32 15e e+ 11 (248 2 ) e e+ 18 (248 2 ) e e-1 11 Accurac test of the ciif scheme with r 1 = r 2 = 1 (248 2 ) e e+ - (248 2 ) e e-1 21 (248 2 ) 64 16e e (248 2 ) e e Accurac test of the ciif scheme with r 1 = r 2 = 2 (248 2 ) e e+ - (248 2 ) e e-1 28 (248 2 ) e e (248 2 ) e e Table 3 Errors and convergence rates at the final time T of Eample 3 (2D semilinear case) with Dirichlet boundar condition and Neumann boundar condition respectivel solutions with such eact solution From Table 4, the second order convergence rate can be again observed as epected 5 Conclusions In high spatial dimensions, the compact representation of integration factor approach was found to be ver efficient for solving sstems involving high-order spatial derivatives and reactions with drasticall different time scales, which in general demand temporal schemes with severe stabilit constraints Unlike the non-compact form, it is difficult to appl ciif directl to handle diffusion terms with cross derivatives In this paper, we have developed a ciif method for solving a class of semilinear fourth-order parabolic equations Because of such representation, computing eponentials of large matrices is reduced to the calculation of eponentials of matrices of significantl smaller sizes, in which the dimension of discretized matrices is the same as one dimensional case and the stabilit condition and computational savings and storage are similar to the original ciif for

20 1686 LILI JU, XINFENG LIU AND WEI LENG (N N ) N t L 2 Error Order L Error Order Dirichlet Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = 1 (248 2 ) e e+ - (248 2 ) e e+ 188 (248 2 ) 64 31e e-1 22 (248 2 ) e e-2 21 Neumann Boundar Condition Accurac test of the ciif scheme with r 1 = r 2 = 1 (248 2 ) e e+ - (248 2 ) e e-1 2 (248 2 ) e e (248 2 ) e e Table 4 Errors and convergence rates at the final time T = 1 of Eample 4 (2D sstem case) with Dirichlet boundar condition and Neumann boundar condition respectivel the second-order problem In addition, the direct and eplicit incorporation of inhomogeneous boundar conditions into the ciif is also proposed Although compact representation has been presented onl in the contet of implicit integration factor methods for semilinear fourth-order parabolic equations, such approach can easil be applied to other integration factor or eponential difference methods Other tpe of equations of high-order derivatives, (eg Cahn-Hilliard equations of fourth-order derivatives) ma also potentiall be handled using the approach for better efficienc To better deal with high spatial dimensions, one ma incorporate the sparse grid [23] into the compact representation technique The fleibilit of compact representation allows either direct calculation of the eponentials of matrices or using Krlov subspace [4, 1, 22] for non-constant diffusion coefficients to compute their eponential matri-vector multiplications for saving further in storages and cost In addition, the presented approach based on the finite difference framework for spatial discretization could also be etended to other discretization methods such as finite volume [8, 11, 17] or spectral methods [3, 24] Overall, the compact representation along with integration factor methods provides an efficient approach for solving a wide range of problems arising from biological and phsical applications Given to its effectiveness in implementation and good stabilit conditions, the method is ver desirable to be incorporated with local adaptive mesh refinement [1, 2, 18], which will also be further eplored in the future work REFERENCES [1] M Berger and P Colella, Local adaptive mesh refinement for shock hdrodnamics, Journal of Computational Phsics, 82 (1989), [2] M Berger and J Oliger, Adaptive mesh refinement for hperbolic partial differential equations, Journal of Computational Phsics, 53 (1984), [3] E O Brigham, The Fast Fourier Transform and its Applications, Prentice Hall, 1988 [4] S Chen and Y-T Zhang, Krlov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: Application to discontinuous Galerkin methods, Journal of Computational Phsics, 23 (211), [5] S M Co and P C Matthews, Eponential time differencing for stiff sstems, Journal of Computational Phsics, 176 (22), [6] Q Du and W Zhu, Stabilit analsis and applications of the eponential time differencing schemes, Journal of Computational Mathematics, 22 (24), 2 29

21 CIIF METHODS FOR SEMILINEAR FOURTH-ORDER PARABOLIC EQUATIONS 1687 [7] Q Du and W Zhu, Modified eponential time differencing schemes: Analsis and applications, BIT Numerical Mathematics, 45 (25), [8] R Emard, T Gallouët and R Herbin, Finite volume methods, Handbook of Numerical Analsis, 7 (2), [9] B Gustafsson, H-O Kreiss and J Oliger, Time Dependent Problems and Difference Methods, volume 67 Wile New York, 1995 [1] M Hochbruck and C Lubich, On krlov subspace approimations to the matri eponential operator, SIAM Journal on Numerical Analsis, 34 (1997), [11] A Jameson, W Schmidt and E Turkel, Numerical Solutions of the Euler Equations b Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes, The 14th AIAA Fluid and Plasma Dnamics Conference, 1981 [12] G-S Jiang and C-W Shu, Efficient implementation of weighted ENO schemes, Journal of Computational Phsics, 126 (1996), [13] L Ju, J Zhang, L Zhu and Q Du, Fast Eplicit Integration Factor Methods for Semilinear Parabolic Equations, Journal of Scientific Computing, 214 [14] A-K Kassam and L N Trefethen, Fourth-order time stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26 (25), [15] B Kleefeld, A Khaliq and B Wade, An ETD Crank-Nicolson method for reaction-diffusion sstems, Numerical Methods for Partial Differential Equations, 28 (212), [16] S Krogstad, Generalized integrating factor methods for stiff PDEs, Journal of Computational Phsics, 23 (25), [17] R LeVeque, Numerical Methods for Conservation Laws, Birkhauser, 1992 [18] X Liu and Q Nie, Compact integration factor methods for comple domains and adaptive mesh refinement, Journal of computational phsics, 229 (21), [19] X-D Liu, S Osher and T Chan, Weighted essentiall non-oscillator schemes, Journal of Computational Phsics, 115 (1994), [2] Q Nie, F Wan, Y-T Zhang and X Liu, Compact integration factor methods in high spatial dimensions, Journal of Computational Phsics, 277 (28), [21] Q Nie, Y-T Zhang and R Zhao, Efficient semi-implicit schemes for stiff sstems, Journal of Computational Phsics, 214 (26), [22] Y Saad, Analsis of some krlov subspace approimations to the matri eponential operator, SIAM Journal on Numerical Analsis, 29 (1992), [23] J Shen and H Yu, Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems, SIAM Journal on Scientific Computing, 32 (21), [24] C Van Loan, Computational Frameworks for the Fast Fourier Transform, volume 1 SIAM, 1992 [25] A Wiegmann, Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds, Lawrence Berkele National Laborator, Paper LBNL-43565, 1999 [26] S Zhao, J Ovadia, X Liu, Y Zhang and Q Nie, Operator splitting implicit integration factor methods for stiff reaction diffusion advection sstems, Journal of Computational Phsics, 23 (211), Received November 213; revised Januar 214 address: ju@mathscedu address: fliu@mathscedu address: wleng@lsecccaccn

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