Spectral Methods for Time-dependent Variable-coefficient PDE Based on Block Gaussian Quadrature

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1 Spectrl Methods for Time-dependent Vrible-coefficient PDE Bsed on Block Gussin Qudrture Jmes V. Lmbers Abstrct Block Krylov subspce spectrl (KSS) methods re best-of-both-worlds compromise between explicit nd implicit time-stepping methods for vrible-coefficient PDE, in tht they combine the efficiency of explicit methods nd the stbility of implicit methods, while lso chieving spectrl ccurcy in spce nd high-order ccurcy in time. Block KSS methods compute ech Fourier coefficient of the solution using techniques developed by Gene Golub nd Gérrd Meurnt for pproximting elements of functions of mtrices by block Gussin qudrture in the spectrl, rther thn physicl, domin. This pper describes how block KSS methods cn be pplied to vriety of equtions, nd lso demonstrtes their superiority, in terms of ccurcy nd efficiency, to other Krylov subspce methods in the literture. Keywords: spectrl methods, Gussin qudrture, block Lnczos method, Mxwell s eqution, het eqution 1 Introduction In [11 clss of methods, clled block Krylov subspce spectrl (KSS) methods, ws introduced for the purpose of solving prbolic vrible-coefficient PDE. These methods re bsed on techniques developed by Golub nd Meurnt in [4 for pproximting elements of function of mtrix by Gussin qudrture in the spectrl domin. In [12, these methods were generlized to the secondorder wve eqution, for which these methods hve exhibited even higher-order ccurcy. It hs been shown in these references tht KSS methods, by employing different pproximtions of the solution opertor for ech Fourier coefficient of the solution, chieve higher-order ccurcy in time thn other Krylov subspce methods (see, for exmple, [9) for stiff systems of ODE, nd they re lso quite stble, considering tht they re Submitted Februry 19, The University of Southern Mississippi, Deprtment of Mthemtics, Httiesburg, MS USA Tel/Fx: /5818 Emil: Jmes.Lmbers@usm.edu explicit methods. They re lso effective for solving systems of coupled equtions, such s Mxwell s equtions [16. In this pper, we review block KSS methods, s pplied to vrious types of PDE, nd compre their performnce to other Krylov subspce methods from the literture. Section 2 reviews the min properties of block KSS methods, s pplied to the prbolic problems for which they were designed. Section 3 discusses implementtion detils, nd demonstrtes why KSS methods need to explicitly generte only one Krylov subspce, lthough informtion from severl is used. In Section 4, we discuss modifictions tht must be mde to block KSS methods in order to pply them to systems of coupled wve equtions, such s Mxwell s equtions. Numericl results re presented in Section 5, nd conclusions re stted in Section 6. 2 Krylov Subspce Spectrl Methods We first review block KSS methods, which re esier to describe for prbolic problems. Let S(t) = exp[ Lt represent the exct solution opertor of the problem u t + Lu = 0, t > 0, (1) with pproprite initil conditions nd periodic boundry conditions. The opertor L is second-order, self-djoint, positive definite differentil opertor. Let, denote the stndrd inner product of functions defined on [0, 2π. Block Krylov subspce spectrl methods, introduced in [11, use Gussin qudrture on the spectrl domin to compute the Fourier coefficients of the solution. These methods re time-stepping lgorithms tht compute the solution t time t 1, t 2,..., where t n = nδt for some choice of Δt. Given the computed solution u(x, t n ) t time t n, the solution t time t n+1 is computed by pproximting the Fourier coefficients tht would be obtined by pplying the exct solution opertor to u(x, t n ), 1 ˆu(ω, t n+1 ) = e iωx, S(Δt) u(x, t n ). (2) 2π

2 In [4 Golub nd Meurnt describe method for computing quntities of the form u T f(a)v, (3) where u nd v re N-vectors, A is n N N symmetric positive definite mtrix, nd f is smooth function. Our gol is to pply this method with A = L N where L N is spectrl discretiztion of L, f(λ) = exp( λt) for some t, nd the vectors u nd v re obtined from ê ω nd u n 1, where ê ω is discretiztion of 2π e iωx nd u n is the pproximte solution t time t n, evluted on n N-point uniform grid. The bsic ide is s follows: since the mtrix A is symmetric positive definite, it hs rel eigenvlues b = λ 1 λ 2 λ N = > 0, (4) nd corresponding orthogonl eigenvectors q j, j = 1,..., N. Therefore, the quntity (3) cn be rewritten s N u T f(a)v = f(λ j )u T q j q T j v. (5) which cn lso be viewed s Riemnn-Stieltjes integrl u T f(a)v = I[f = f(λ) dα(λ). (6) As discussed in [4, the integrl I[f cn be pproximted using Gussin qudrture rules, which yields n pproximtion of the form I[f = K w j f(λ j ) + R[f, (7) where the nodes λ j, j = 1,..., K, s well s the weights w j, j = 1,..., K, cn be obtined using the symmetric Lnczos lgorithm if u = v, nd the unsymmetric Lnczos lgorithm if u = v (see [7). In the cse u = v, there is possibility tht the weights my not be positive, which destbilizes the qudrture rule (see [1 for detils). Insted, we consider [ u v T f(a) [ u v, (8) which results in the 2 2 mtrix [ u f(λ) dμ(λ) = T f(a)u u T f(a)v v T f(a)u v T f(a)v, (9) where μ(λ) is 2 2 mtrix function of λ, ech entry of which is mesure of the form α(λ) from (6). In [4 Golub nd Meurnt showed how block method cn be used to generte qudrture formuls. We will describe this process here in more detil. The integrl f(λ) dμ(λ) is now 2 2 symmetric mtrix nd the most generl K-node qudrture formul is of the form f(λ) dμ(λ) = K W j f(t j )W j + error, (10) with T j nd W j being symmetric 2 2 mtrices. By digonlizing ech T j, we obtin the simpler formul f(λ) dμ(λ) = 2K f(λ j )v j v T j + error, (11) where, for ech j, λ j is sclr nd v j is 2-vector. Ech node λ j is n eigenvlue of the mtrix M 1 B1 T B 1 M 2 B T 2 T K = , (12) B K 2 M K 1 BK 1 T B K 1 M K which is block-tringulr mtrix of order 2K. The vector v j consists of the first two elements of the corresponding normlized eigenvector. To compute the mtrices M j nd B j, we use the block Lnczos lgorithm, which ws proposed by Golub nd Underwood in [6. We re now redy to describe block KSS methods. For ech wve number ω = N/2 + 1,..., N/2, we define R 0 (ω) = [ ê ω u n nd compute the QR fctoriztion R 0 (ω) = X 1 (ω)b 0 (ω). We then crry out block Lnczos itertion, pplied to the discretized opertor L N, to obtin block tridigonl mtrix T K (ω) of the form (12), where ech entry is function of ω. Then, we cn express ech Fourier coefficient of the pproximte solution t the next time step s [û n+1 ω = [ B0 H E12 H exp[ T K (ω)δte 12 B 0 (13) where E 12 = [ e 1 e 2. The computtion of (13) consists of computing the eigenvlues nd eigenvectors of T K (ω) in order to obtin the nodes nd weights for Gussin qudrture, s described erlier. This lgorithm hs locl temporl ccurcy O(Δt 2K 1 ) [11. Furthermore, block KSS methods re more ccurte thn the originl KSS methods described in [14, even though they hve the sme order of ccurcy, becuse the solution u n plys greter role in the determintion of the qudrture nodes. They re lso more effective 12

3 for problems with oscilltory or discontinuous coefficients [11. Block KSS methods re even more ccurte for the second-order wve eqution, for which block Lnczos itertion is used to compute both the solution nd its time derivtive. In [12, Theorem 6, it is shown tht when the leding coefficient is constnt nd the coefficient q(x) is bndlimited, the 1-node KSS method, which hs secondorder ccurcy in time, is lso unconditionlly stble. In generl, s shown in [12, the locl temporl error is O(Δt 4K 2 ) when K block Gussin nodes re used. 3 Implementtion KSS methods compute Jcobi mtrix corresponding to ech Fourier coefficient, in contrst to trditionl Krylov subspce methods tht normlly use only single Krylov subspce generted by the initil dt or the solution from the previous time step. While it would pper tht KSS methods incur substntil mount of dditionl computtionl expense, tht is not ctully the cse, becuse nerly ll of the Krylov subspces tht they compute re closely relted by the wve number ω, in the 1-D cse, or ω = (ω 1, ω 2,..., ω n ) in the n-d cse. In fct, the only Krylov subspce tht is explicitly computed is the one generted by the solution from the previous time step, of dimension (K + 1), where K is the number of block Gussin qudrture nodes. In ddition, the verges of the coefficients of L j, for j = 0, 1, 2,..., 2K 1, re required, where L is the sptil differentil opertor. When the coefficients of L re independent of time, these cn be computed once, during preprocessing step. This computtion cn be crried out in O(N log N) opertions using symbolic clculus [13, 15. With these considertions, the lgorithm for single time step of 1-node block KSS method for solving (1), where Lu = pu xx + q(x)u, with pproprite initil conditions nd periodic boundry conditions, is s follows. We denote the verge of function f(x) on [0, 2π by f, nd the computed solution t time t n by u n. ˆu n = fft(u n ), v = Lu n, ˆv = fft(v) for ech ω do α 1 = pω 2 + q (in preprocessing step) β 1 = ˆv(ω) α 1ˆu n (ω) α 2 = u n, v 2 Re [ˆu n (ω)v(ω) + α 1 u n (ω) 2 e ω = [ u [ n, u n ˆu n (ω) 2 1/2 α 1 β 1 /e ω T ω = β 1 /e ω α 2 /e 2 ω ˆu n+1 (ω) = [e TωΔt 11ˆu n (ω) + [e TωΔt 12 e ω end u n+1 = ifft(ˆu n+1 ) It should be noted tht for prbolic problem such s (1), the loop over ω only needs to ccount for nonnegligible Fourier coefficients of the solution, which re reltively few due to the smoothness of solutions to such problems. 4 Appliction to Mxwell s Equtions We consider Mxwell s eqution on the cube [0, 2π 3, with periodic boundry conditions. Assuming nonconductive mteril with no losses, we hve div Ê = 0, div Ĥ = 0, (14) curl Ê = μ Ĥ t, curl Ĥ = ε Ê t, (15) where Ê, Ĥ re the vectors of the electric nd mgnetic fields, nd ε, μ re the electric permittivity nd mgnetic permebility, respectively. Tking the curl of both sides of (15) yields με 2 Ê t 2 = ΔÊ + μ 1 curl Ê μ, (16) με 2 Ĥ t 2 = ΔĤ + ε 1 curl Ĥ ε. (17) In this section, we discuss generliztions tht must be mde to block KSS methods in order to pply them to non-self-djoint system of coupled equtions such s (16). Additionl detils re given in [16. First, we consider the following 1-D problem, 2 u + Lu = 0, t > 0, (18) t2 with pproprite initil conditions, nd periodic boundry conditions, where u : [0, 2π [0, ) R n for n > 1, nd L(x, D) is n n n mtrix where the (i, j) entry is n differentil opertor L ij (x, D) of the form m ij L ij (x, D)u(x) = ij μ (x)d μ u, μ=0 D = d dx, (19) with sptilly vrying coefficients ij μ, μ = 0, 1,..., m ij. Generliztion of KSS methods to system of the form (18) cn proceed s follows. For i, j = 1,..., n, let L ij (D) be the constnt-coefficient opertor obtined by verging the coefficients of L ij (x, D) over [0, 2π. Then, for ech wve number ω, we define L(ω) be the mtrix with entries L ij (ω), i.e., the symbols of L ij (D) evluted t ω.

4 Next, we compute the spectrl decomposition of L(ω) for ech ω. For j = 1,..., n, let q j (ω) be the Schur vectors of L(ω). Then, we define our test nd tril functions by φ j,ω (x) = q j (ω) e iωx. For Mxwell s equtions, the mtrix A N tht discretizes the opertor AÊ = 1 ) (ΔÊ με + μ 1 curl Ê μ is not symmetric, nd for ech coefficient of the solution, the resulting qudrture nodes λ j, j = 1,..., 2K, from (11) re now complex nd must be obtined by strightforwrd modifiction of block Lnczos itertion for unsymmetric mtrices. 5 Numericl Results In this section, we compre the performnce of block KSS methods with vrious methods bsed on exponentil integrtors [8, 10, Prbolic Problems We first consider 1-D prbolic problem of the form (1), where the differentil opertor L is defined by Lu(x) = pu (x) + q(x)u(x), where p 0.4 nd q(x) cos x 0.02 sin x cos 2x sin 2x cos 3x is constructed so s to hve the smoothness of function with three continuous derivtives, s is the initil dt u(x, 0). Periodic boundry conditions re imposed. Since the exct solution is not vilble, the error is estimted by tking the l 2 -norm of the reltive difference between ech solution, nd tht of solution computed using smller time step Δt = 1/64 nd the mximum number of grid points. The results re shown in Figures 1 nd 2. As the number of grid points is doubled, only the block KSS method shows n improvement in ccurcy; the preconditioned Lnczos method exhibits slight degrdtion in performnce, while the explicit fourth-order exponentil integrtor-bsed method requires tht the time step be reduced by fctor of 4 before it cn deliver the expected order of convergence; similr behvior ws demonstrted for n explicit 3rd-order method from [9 in [14. The preconditioned Lnczos method requires 8 Lnczos itertions to mtch the ccurcy of block KSS method tht uses only 2. On the other hnd, the block KSS method incurs dditionl expense due to (1) the computtion of the moments of L, for ech Fourier coefficient, nd (2) the exponentition of seprte Jcobi mtrices for ech Fourier coefficient. These expenses re mitigted by the fct tht the first tkes plce once, during preprocessing stge, nd both tsks require n mount of work tht is proportionl not to the number of grid points, but to the number of non-negligible Fourier coefficients of the solution. We solve this problem using the following methods: A 2-node block KSS method. Ech time step requires construction of Krylov subspce of dimension 3 generted by the solution, nd the coefficients of L 2 nd L 3 re computed during preprocessing step. A preconditioned Lnczos itertion for pproximting e τa v, introduced in [17 for pproximting the mtrix exponentil of sectoril opertors, nd dpted in [19 for efficient ppliction to the solution of prbolic PDE. In this pproch, Lnczos itertion is pplied to (I+hA) 1, where h is prmeter, in order to obtin restricted rtionl pproximtion of the mtrix exponentil. We use m = 4 nd m = 8 Lnczos itertions, nd choose h = Δt/10, s in [19. A method bsed on exponentil integrtors, from [8, tht is of order 3 when the Jcobin is pproximted to within O(Δt). We use m = 8 Lnczos itertions. Figure 1: Estimtes of reltive error t t = 0.1 in solutions of (1) computed using preconditioned exponentil integrtor [19 with 4 nd 8 Lnczos itertions, nd 2- node block KSS method. All methods compute solutions on n N-point grid, with time step Δt, for vrious vlues of N nd Δt.

5 in these experiments, we use m = 2 Lnczos itertions to pproximte the Jcobin. It should be noted tht when m is incresed, even to substntil degree, the results re negligibly ffected. Figure 3 demonstrtes the convergence behvior for both methods. At both sptil resolutions, the block KSS method exhibits pproximtely 6th-order ccurcy in time s Δt decreses, except tht for N = 16, the sptil error rising from trunction of Fourier series is significnt enough tht the overll error fils to decrese below the level chieved t Δt = 1/8. For N = 32, the solution is sufficiently resolved in spce, nd the order of overgence s Δt 0 is pproximtely 6.1. Figure 2: Estimtes of reltive error t t = 0.1 in solutions of (1) computed using 4th-order method bsed on n exponentil integrtor [10 with 8 Lnczos itertions, nd 2-node block KSS method. Both methods compute solutions on n N-point grid, with time step Δt, for vrious vlues of N nd Δt. 5.2 Mxwell s Equtions We now pply 2-node block KSS method to (16), with initil conditions Ê Ê(x, y, z, 0) = F(x, y, z), (x, y, z, 0) = G(x, y, z), t (20) with periodic boundry conditions. The coefficients μ nd ε re given by μ(x, y, z) = cos z cos y sin y cos(y + z) cos(y z) cos x cos(x z) sin(x z) cos(x + y) cos(x y), ε(x, y, z) = cos z cos y cos(y + z) cos x cos(x z) cos(x + y) cos(x y). The components of F nd G re generted in similr fshion, except tht the x- nd z-components re zero. We use block KSS method tht uses K = 2 block qudrture nodes per coefficient in the bsis described in Section 4, tht is 6th-order ccurte in time, nd cosine method bsed on Gutschi-type exponentil integrtor [8, 10. This method is second-order in time, nd We lso note tht incresing the resolution does not pose ny difficulty from stbility point of view. Unlike explicit finite-difference schemes tht re constrined by CFL condition, KSS methods do not require reduction in the time step to offset reduction in the sptil step in order to mintin boundedness of the solution, becuse their domin of dependence includes the entire sptil domin for ny Δt. The Gutschi-type exponentil integrtor method is second-order ccurte, s expected, nd delivers nerly identicl results for both sptil resolutions, but even with Krylov subspce of much higher dimension thn tht used in the block KSS method, it is only ble to chieve t most second-order ccurcy, wheres block KSS method, using Krylov subspce of dimension 3, chieves sixth-order ccurcy. This is due to the incorportion of the moments of the sptil differentil opertor into the computtion, nd the use of Gussin qudrture rules specificlly tilored to ech Fourier coefficient. 6 Summry nd Future Work We hve demonstrted tht block KSS methods cn be pplied to Mxwell s equtions with smoothly vrying coefficients, by pproprite generliztion of their ppliction to the sclr second-order wve eqution, in wy tht preserves the order of ccurcy chieved for the wve eqution. Furthermore, it hs been demonstrted tht while trditionl Krylov subspce methods bsed on exponentil integrtors re most effective for prbolic problems, especilly when ided by preconditioning s in [19, KSS methods perform best when pplied to hyperbolic problems, in view of their much higher order of ccurcy. Future work will extend the pproch described in this pper to more relistic pplictions involving Mxwell s equtions by using symbol modifiction to efficiently implement perfectly mtched lyers (see [2), nd vrious techniques (see [3, 18) to effectively hndle discontinuous coefficients.

6 [9 Hochbruck, M., Lubich, C.: On Krylov Subspce Approximtions to the Mtrix Exponentil Opertor. SIAM Journl of Numericl Anlysis 34 (1997) [10 Hochbruck, M., Lubich, C., Selhofer, H.: Exponentil Integrtors for Lrge Systems of Differentil Equtions. SIAM Journl of Scientific Computing 19 (1998) [11 Lmbers, J. V.: Enhncement of Krylov Subspce Spectrl Methods by Block Lnczos Itertion. Electronic Trnsctions on Numericl Anlysis 31 (2008) Figure 3: Estimtes of reltive error t t = 1 in solutions of (16), (20) computed using cosine method bsed on Gutschi-type exponentil integrtor [8, 10 with 2 Lnczos itertions, nd 2-node block KSS method. Both methods compute solutions on n N 3 -point grid, with time step Δt, for vrious vlues of N nd Δt. References [1 Atkinson, K.: An Introduction to Numericl Anlysis, 2nd Ed. Wiley (1989) [2 Berenger, J,: A perfectly mtched lyer for the bsorption of electromgnetic wves. J. Comp. Phys. 114 (1994) [3 Gelb, A., Tnner, J.: Robust Reprojection Methods for the Resolution of the Gibbs Phenomenon. Appl. Comput. Hrmon. Anl. 20 (2006) [4 Golub, G. H., Meurnt, G.: Mtrices, Moments nd Qudrture. Proceedings of the 15th Dundee Conference, June-July 1993, Griffiths, D. F., Wtson, G. A. (eds.), Longmn Scientific & Technicl (1994) [5 Golub, G. H., Gutknecht, M. H.: Modified Moments for Indefinite Weight Functions. Numerische Mthemtik 57 (1989) [6 Golub, G. H., Underwood, R.: The block Lnczos method for computing eigenvlues. Mthemticl Softwre III, J. Rice Ed., (1977) [7 Golub, G. H, Welsch, J.: Clcultion of Guss Qudrture Rules. Mth. Comp. 23 (1969) [8 Hochbruck, M., Lubich, C.: A Gutschi-type method for oscilltory second-order differentil equtions, Numerische Mthemtik 83 (1999) [12 Lmbers, J. V.: An Explicit, Stble, High-Order Spectrl Method for the Wve Eqution Bsed on Block Gussin Qudrture. IAENG Journl of Applied Mthemtics 38 (2008) [13 Lmbers, J. V.: Krylov Subspce Spectrl Methods for the Time-Dependent Schrödinger Eqution with Non-Smooth Potentils. Numericl Algorithms 51 (2009) [14 Lmbers, J. V.: Krylov Subspce Spectrl Methods for Vrible-Coefficient Initil-Boundry Vlue Problems. Electronic Trnsctions on Numericl Anlysis 20 (2005) [15 Lmbers, J. V.: Prcticl Implementtion of Krylov Subspce Spectrl Methods. Journl of Scientific Computing 32 (2007) [16 Lmbers, J. V.: A Spectrl Time-Domin Method for Computtionl Electrodynmics. Advnces in Applied Mthemtics nd Mechnics 1 (2009) [17 Moret, I., Novti, P.: RD-rtionl pproximtion of the mtrix exponentil opertor. BIT 44 (2004) [18 Vllius, T., Honknen, M.: Reformultion of the Fourier nodl method with dptive sptil resolution: ppliction to multilevel profiles. Opt. Expr. 10(1) (2002) [19 vn den Eshof, J., Hochbruck, M.: Preconditioning Lnczos pproximtions to the mtrix exponentil. SIAM Journl of Scientific Computing 27 (2006)