Stability Analysis of the Matrix-Free Linearly Implicit 2 Euler Method 3 UNCORRECTED PROOF

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1 1 Stability Analysis of the Matrix-Free Linearly Iplicit 2 Euler Method 3 Adrian Sandu 1 andaikst-cyr Coputational Science Laboratory, Departent of Coputer Science, Virginia 5 Polytechnic Institute, Blacksburg, VA, 24060, USA. E-ail sandu@cs.vt.edu 6 2 National Center for Atospheric Research, 1850 Table Mesa Drive, Boulder CO, E-ail aik@ucar.edu. 8 Suary. Iplicit tie stepping ethods are useful for the siulation of large scale PDE 9 systes because they avoid the tie step liitations iposed by explicit stability conditions. 10 To alleviate the challenges posed by coputational and eory constraints, any applica- 11 tions solve the resulting linear systes by iterative ethods where the Jacobian-vector prod- 12 ucts are approxiated by finite differences. This paper explains the relation between a linearly 13 iplicit Euler ethod, solved using a Jacobian-free Krylov ethod, and explicit Runge-Kutta 14 ethods. The case with preconditioning is equivalent to a Rosenbrock-W ethod where the 15 approxiate Jacobian, inverted at each stage, corresponds directly to the preconditioner. The 16 accuracy of the resulting Runge-Kutta ethods can be controlled by constraining the Krylov 17 solution. Nuerical experients confir the theoretical findings Introduction 19 Large systes of tie dependent partial differential equations PDEs), arising in 20 ulti-physics siulations, are often discretized using the ethod of lines approach. 21 The independent tie and space nuerical schees allow the coupling of ultiple 22 physics odules, and provide axiu flexibility in choosing appropriate algo- 23 riths. After the sei-discretization in space the syste of PDEs is reduced to a 24 syste of ordinary differential equations ODEs) 25 y = f y), t 0 t t f, yt 0 )=y 0. 1) Here yt) R d is the solution vector and y 0 the initial condition. We denote the 26 Jacobian of the ODE function by Jy) = f y y) R d d, and the identity atrix by 27 I R d d. 28 Stability requireents e.g., the CFL condition for discretized hyperbolic PDEs) 29 liit the tie steps allowable by explicit tie discretizations of 1). When the fastest 30 tie scales in the syste 1) are short, e.g., in the presence of fast waves, the stability 31 condition iposes tie steps uch saller then those required to achieve the target 32 R. Bank et al. eds.), Doain Decoposition Methods in Science and Engineering XX, Lecture Notes in Coputational Science and Engineering 91, DOI / , Springer-Verlag Berlin Heidelberg 2013 Page 427

2 Adrian Sandu and Aik St-Cyr accuracy. The step size liitation by linear stability conditions is referred to as stiff- 33 ness. In order to overcoe this coputational inefficiency, it is desirable to use i- 34 plicit, unconditionally stable discretizations which allow arbitrarily large tie steps 35 [2]. Iplicit ethods have a high cost per step due to the need to solve a non)linear 36 syste of equations. 37 To reduce the coputational and eory costs of direct linear syste solvers, 38 and to aid parallelization, iterative Krylov space ethods are eployed. Further- 39 ore, atrix-free ipleentations approxiate Jacobian vector products by finite 40 differences [4]. This approach avoids additional coding for the Jacobian, preserves 41 the parallel scalability of the explicit odel, and has becoe popular in any appli- 42 cations, e.g., [1, 5, 6]. The hope is that the properties of the iplicit tie discretiza- 43 tion reain unaltered, provided that the iterative solutions are carried out to sufficient 44 accuracy. We show here that the atrix-free approach does alter the properties of the 45 underlying iplicit tie stepping ethod. 46 This study treats a linearly iplicit ethod, together with the Krylov subspace 47 iterations for solving the linear syste, as a single nuerical schee. The analysis 48 reveals that atrix-free ipleentations of linearly iplicit ethods are equivalent 49 to explicit Runge Kutta ethods. Consequently, the unconditional stability property 50 of the base ethod is lost. When preconditioning is used, the atrix-free iplicit 51 ethods are equivalent to Rosenbrock-W ROS-W) ethods where the approxiate 52 Jacobians correspond directly to the preconditioners The Matrix-Free Linearly Iplicit Euler Method 54 Consider the linearly iplicit Euler LIE) ethod applied to 1) 55 I ΔtJyn ) ) w = f y n ), y n+1 = y n + Δt w. 2) When the linear syste is solved exactly odulo roundoff errors) by LU factoriza- 56 tion the ethod 2) is unconditionally stable, and thus suitable for the solution of 57 stiff systes. For any PDEs sei-discretized in the ethod of lines fraework, 58 however, the diension of the linear syste 2) is very large, and the coputational 59 and eory costs associated with a direct solution are prohibitive. Moreover, the 60 construction of the explicit Jacobian atrix J is difficult when the space discretiza- 61 tion is based on a doain decoposition approach. To alleviate these probles, a 62 popular approach is to solve 2) by atrix-free iterative ethods. We seek to analyze 63 the ipact that this approxiate solutions have on the stability and accuracy of the 64 iplicit tie stepping schee. Our approach is to treat the original discretization 65 2) together with the iterations as a single nuerical ethod applied to solve the 66 ODE 1). 67 To be specific, we solve the linear syste in 2) by a Krylov space ethod. The 68 initial guess is y n+1 = y n, i.e., w = 0. After iterations the following -diensional 69 Krylov space is built: 70 { K = span f y n ),..., I ΔtJy n ) ) } 1 f yn ). Page 428

3 Stability Analysis of the Matrix-Free Linearly Iplicit Euler Method In the atrix-free approach, the basis is constructed recursively and the Jacobian- 71 vector products are approxiated by finite differences 72 l i = l i 1 Δt ε 1 f y n + ε l i 1 )+Δt ε 1 l 1, i = 2,...,. 3) We assue that the sae scaling factor ε is used to copute the finite differences in 73 all iterations. The analysis can be easily extended to the case where a different ε is 74 used in each iteration.) Denote 75 k 1 = f y n ); k i = f y n + εl i 1 ), i = 2,,. 4) The recurrence 3) can be expressed in ters of k i as: 76 k i = f y n + Δt Δt 1 ε +i 2) ) i 1 k 1 Δt k j ), i = 2,...,. 5) The solution w = α i l i K can be expressed in ters of k i s: 77 ) ) w = α i + Δt ε 1 i 1)α i k 1 Δt ε 1 α j k i. 6) i=2 j=2 i=2 j=i Equations 5)and6), together with the relation y n+1 = y n +Δt w, are copared with 78 the -stage explicit Runge Kutta ERK) ethod [3] 79 i 1 k i = f y n + a ij k j ), i = 1,...,; y n+1 = y n + Δt b i k i. j=1 The coparison reveals the following. 80 Theore 1. The atrix-free LIE 2) ethod is equivalent to an explicit Runge Kutta 81 ethod. The nuber of Krylov iterations defines the nuber of Runge Kutta stages. 82 Equations 5)and6) define the coefficients of the ERK ethod: 83 a i,1 = Δt 1 ε +i 2); a i, j = 1, for i = 2,,, j = 2,,i 1; b 1 = j + Δt ε j=1α 1 j 1)α j ; b i = Δt ε 1 α j, i = 2,...,. j=2 j=i 2.1 Stability Considerations 84 The solution of the linear syste under the initial guess w = 0) is part of the Krylov 85 space K and can be represented by a atrix polynoial 86 w = p 1 I Δt Jyn ) ) f y n ). 87 The atrix-free LIE ethod applied to the Dahlquist test proble y = λ y, y0)=1, 88 gives the following solution: 89 y n+1 = y n + Δtw=1 + zp 1 1 z)) y n = Rz)y n, 90 with z = Δt λ. The stability function of the equivalent ERK ethod is the degree 91 polynoial Rz)=1 + zp 1 1 z). 92 Page 429

4 Adrian Sandu and Aik St-Cyr Theore 2. The stability region of the LIE ethod, with a Krylov atrix-free linear 93 solver, is necessarily finite. The unconditional stability of the original LIE ethod is 94 lost. 95 Siilar considerations hold for Krylov space ethods that use an orthogonal basis 96 of the Krylov space, built by Arnoldi iterations [7] Accuracy Considerations 98 The ethod accuracy is difficult to assess, as the coefficients depend on the tie 99 step. The relation between the finite difference scaling factor ε and the tie step Δt 100 is iportant in deterining accuracy. 101 Assue that the finite difference scaling factor is a constant fraction of the tie 102 step, ε/δt = const. This is a reasonable assuption: in order to increase accuracy 103 one decreases both Δt, to reduce the truncation error, and ε, to reduce the finite 104 difference error. Of course, for very sall ε the finite difference error becoes again 105 large due to roundoff.) Also assue that the coefficients α 1,...,α do not depend 106 on ε or Δt. 107 In this case the accuracy can be assessed using the classical approach. The order 108 conditions depend on the Krylov space coefficients α as follows: 109 Order 1: Order 2: b i = j=1 b i c i = α j = 1, i=2 7a) i 1)α i = b) Neither condition 7a) nor 7b) are autoatically satisfied by the Krylov iterative 110 ethods. In particular, 111 Lea 1. The first order accuracy of the atrix-free LIE is not autoatic when 112 ε/δt = const. Additional constraints need to be iposed on the Krylov solution 113 coefficients. 114 Consider now the case where ε is constant does not depend on Δt ). Assue 115 that the coefficients α 1,...,α do not depend on ε or Δt. A necessary condition for 116 the ethod to be accurate of order p is that its stability function approxiates the 117 exponential, Rz)=e z + O z p+1). The stability function does not depend on either 118 ε or Δt. The conditions 7a)and7b) on the Krylov solution coefficients α 1,...,α, 119 which are sufficient when ε = const Δt, see to be necessary in the case ε = const. 120 In the general case the Krylov solution coefficients α 1,...,α do depend on Δt. 121 For Δt 0wehavethatw f y n ) and therefore α 1 1, α 2,α 3, Asyp- 122 totically the condition 7a) holds. Moreover, the nuberof iterations also depends 123 on Δt through the convergence speed. Consequently, it is difficult to extend the clas- 124 sical accuracy analysis to atrix-free linearly iplicit ethods. It sees reasonable, 125 however, to odify the Krylov ethod and ipose at least condition 7a) onthe 126 Krylov coefficients. 127 Page 430

5 Stability Analysis of the Matrix-Free Linearly Iplicit Euler Method 3 Preconditioned Iterations 128 Consider the case where a preconditioner atrix M is used to speed up the iterations. 129 The linear syste 2) becoes 130 M 1 I ΔtJy n ) ) k = M 1 f y n ). 131 The Krylov space constructed in this case is 132 { K = span f y n )..., M 1 I ΔtJy n )) ) } 1 M 1 f y n ). In the atrix-free approach the following basis is constructed recursively 133 l 1 = M 1 f y n ), l i = M 1 l i 1 Δt ε 1 M 1 f y n + ε l i 1 )+Δt ε 1 l 1, i = 2,...,. Denote k 1 = Δt l 1 and k i = Δt l 1 ε l i for i = 2,,. Wehave 134 Mk 1 = Δt fy n ) 8) Mk i = Δt fy n + k 1 k i 1 )+k i 1 k 1, i = 2,...,. Consider, for coparison, a Rosenbrock-W ROW) ethod in the ipleentation- 135 friendly forulation [2, Sect. IV.7] 136 ) ] i 1 i 1 [I Δt γ Ĵ n k i = Δt γ f y n + a ij k j + γ c ij k j, y n+1 = y n + s j=1 j=1 i k i. 9) Here Ĵ n J y n ) is an approxiation of the exact Jacobian at the current step. We 137 identify the ethod coefficients γ = 1and 138 c i,1 = 1; c i,i 1 = 1; a i,1 = 1; a i,i 1 = 1, i = 2,,. Fro the solution w = α i l i = b i k i K we identify the weights 139 b 1 = α 1 Δt 1 + ε 1 α j ; j=2 b i = ε 1 α i, i = 2,...,. The preconditioner defines the Jacobian approxiation in the ROW ethod, 140 M = I Δt γ Ĵ n Ĵ n = Δt 1 I M). Theore 3. The preconditioned atrix-free LIE is equivalent to a linearly-iplicit 141 ROW ethod. The choice of the preconditioner, besides accelerating convergence, 142 iproves the stability of the atrix-free LIE ethod. The preconditioner defines the 143 Jacobian approxiation in the ROW ethod. 144 Note that the general approach can be applied to ROW ethods [2, Sect. IV.7] 145 by solving the linear syste of each stage with an iterative atrix free algorith. 146 The resulting schee is an explicit Runge Kutta ethod or a ROW ethod) with 147 s i stages. 148 Page 431

6 Adrian Sandu and Aik St-Cyr 4 Nuerical Results 149 Consider the one diensional scalar advection-diffusion equation 150 this figure will be printed in b/w u t +au) x = Du xx, ux,t = 0)=u 0 x). 10) A spectral discontinuous Galerkin spatial discretization is used with 20 eleents and 151 polynoials of order 8. The diffusive ter discretization is stabilized using the inter- 152 nal penalty ethod [8]. The LIE tie stepping is used with the atrix-free GMRES 153 solver [7]. 154 Iaginary Σ b i =0.60; M=4; CFL=2.05; Diff=1.0e Real a) D=0.001, =4 Iaginary Σ b i =0.94; M=10; CFL=2.05; Diff=1.0e Real b) D=0.001, =10 M=9; LIE Orders: Gauss=0.98 GMRES=0.98 MFree= CFL Error c) D=0:001; =9 Fig. 1. a) andb) The ERK stability regions for different nubers of GMRES iterations. c) The accuracy of the LIE schee using various approaches to invert the Jacobian atrix. The GMRES weights are restricted by 7b) such as to obtain a second order ethod. Advectiondiffusion equation 10), Δt = CFL, ε = 10 6 Δt In Fig.1a, b, the stability regions generated by the GMRES iterations are plotted 155 for a varying nuber of Krylov vectors. The regions grow quickly and encopass 156 the eigenvalues of the discrete advection-diffusion operator. Subsequent iterations 157 iprove solution accuracy but do not iprove linear stability. Additional experi- 158 ents not reported here due to space constraints) reveal that the stability region of 159 the resulting ERK ethod adapts to the eigenvalues of different discrete operators. 160 To verify the analysis in 7), we consider three different ways of coputing the 161 inverse of the linear Jacobian. The first is by Gauss eliination LU), the second 162 uses GMRES with the full Jacobian, and the third eploys atrix-free GMRES it- 163 erations. In the last approach the GMRES coefficients are restricted by 7b) such 164 as to obtain a second order tie discretization ethod. Figure 1c shows the work- 165 precision diagra for these approaches. The Gaussian eliination and traditional 166 GMRES solutions display first order converge, while the constrained GMRES solu- 167 tion displays second order convergence. 168 Gauss GMRES Mat.Free 5 Conclusions 169 Iplicit tie integration ethods are becoing widely used in the the siulation of 170 tie dependent PDEs, as they do not suffer fro CFL stability restrictions. While 171 Page 432

7 Stability Analysis of the Matrix-Free Linearly Iplicit Euler Method iplicit ethods can use uch larger tie steps than explicit ethods, their co- 172 putational cost per step is also higher. The coputational tie is doinated by the 173 solutions of non)linear systes of equations that define each stage of a linearly) 174 iplicit ethod. The iplicit code is ore effective only when the gains in step size 175 offset the extra cost. 176 To reduce the coputational overhead of LU decoposition, to alleviate eory 177 requireents, and to aid parallelization, iterative Krylov space ethods are used 178 to solve the large linear systes. A atrix-free ipleentation approxiates the 179 required Jacobian vector products by finite differences. 180 This paper studies the effect of the atrix-free iterative solutions on the proper- 181 ties of the nuerical integration ethod. The analysis reveals that atrix-free lin- 182 early iplicit ethods can be viewed as explicit Runge Kutta ethods. Their stabil- 183 ity region is finite, and the unconditional stability property of the original iplicit 184 ethod is lost. The equivalent Runge Kutta ethod is nonlinear, in the sense that 185 its weights depend on the tie step and on the stage vectors. This akes the ac- 186 curacy analysis difficult. Order conditions of the equivalent explicit Runge Kutta 187 ethod can be fulfilled by iposing additional conditions on the Krylov solution 188 coefficients. For preconditioned atrix-free iterations the overall tie stepping pro- 189 cess is equivalent to a Rosenbrock-W ethod, where the preconditioner deterines 190 the Jacobian approxiation. Future work will address the effect of a finite nuber 191 of Krylov iterations on the stability and accuracy of the overall schee, in the case 192 where an analytical Jacobian is used. 193 Acknowledgents This work was supported by the NSF projects DMS , CCF , OCI , PetaApps A. Sandu was also supported by NCAR s ASP 195 Faculty Fellowship Progra. 196 Bibliography 197 [1] A. Crivellini and F. Bassi. An iplicit atrix-free Discontinuous Galerkin solver 198 for viscous and turbulent aerodynaic siulations. Coputers and Fluids, ):81 93, [2] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and 201 Differential-Algebraic Probles. 2Ed. Springer-Verlag, [3] E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary Differential Equations 203 I. Nonstiff Probles. Springer-Verlag, Berlin, [4] Ti Kelley. Iterative Methods for Optiization. SIAM, [5] D.A. Knoll and D.E. Keyes. Jacobian-free Newton-Krylov ethods: a survey 206 of approaches and applications. Journal of Coputational Physics, 1932): , [6] H. Luo, J.D. Bau, and R. Lohner. A fast, atrix-free iplicit ethod for co- 209 pressible flows on unstructured grids. In Sixteenth international conference on 210 nuerical ethods in fluid dynaics, volue 515 of Lecture Notes in Physics, 211 pages 73 78, Page 433

8 Adrian Sandu and Aik St-Cyr [7] Y. Saad. Iterative Methods for Sparse Linear Systes. SIAM, 2 edition, [8] K. Shahbazi. An explicit expression for the penalty paraeter of the interior 214 penalty ethod. Journal of Coputational Physics, 205: , Page 434