2003Kluwer Academic Publishers. Printed in the Netherlands.
Multiparameter schemes for evolutionary equations
J.P. Chehabaand B. Costab
aLaboratoire de Maths applis, CNRS FREE 2222, Université de Lille 1, Bât. M2, Villeneuve d’Ascq, and Equipe Analyse Numérique et EDP, Bât. 425, Université Paris XI, Orsay, France
E-mail: [email protected]
bDepartamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro, Cidade Universitária, Ilha do Fundão, Rio de Janeiro, RJ, Brazil
E-mail: [email protected]
Received 26 November 2001; accepted 20 June 2003
Dedicated to Claude Brezinski on the occasion of his 60th birthday
Multiparameter extensions (MP) of (linear and nonlinear) descent methods have been pro- posed for the solution of finite dimensional time independent problems; these new methods are based on a different treatment of several blocks of components of the solution, basically via the substitution of a scalar relaxation by a (suitable) matricial relaxation. Similarly, the Non- linear Galerkin Method (NLG), that stems from the dynamical system theory, propose to apply distinct temporal integration schemes to different sets of data scales when solving dissipative PDEs. In this paper, the algebraic similarity of Richardson iteration and Forward-Euler time integration is extended to new grounds through the expansion of the realm of MP methods to the field of the numerical integration of dissipative PDEs. The separation of the structures is realized by the utilization of hierarchical preconditioners in finite differences, which are con- jugated to a MP temporal integration steeming from NLG theory. Numerical examples of fluid dynamics problems show the improved temporal stability of these new methods as compared to the classical ones.
Keywords: nonlinear Galerkin methods, separation of scales, multiparameter methods, sta- bility, iterative processes
AMS subject classification:65M06, 65M99, 65N10, 65N12, 65N22
1. Introduction
In the classical schemes of numerical analysis, stability and convergence condi- tions are usually chosen regarding extreme situations (control of numerical modes, for instance), being too harsh and restrictive for most of the components of the solution; gen- erally, noa prioriinformation on the solution is exploited for deriving a larger domain of stability. Let us give a simple but very instructive example: consider the numerical solution of a linear system AX = F by the classical Richardson method, A being a
n×npositive definite matrix andX,F two vectors ofRn. It consists in constructing a sequence of type
X(n+1) =X(n)−α
AX(n)−F ,
where the parameterαmust be chosen in the interval]0,2/ρ(A)[to insure convergence.
Now, assume in addition thatA is symmetric. Then, rewriting the last system in the eigenvector basis ofA, one obtain
Y(n+1)=Y(n)−α
DY(n)−G , where we have set
D=P−1AP , Y =P−1X and G=P−1F,
Dbeing the diagonal matrix with the eigenvalues(λi)Ni=1ofAas entries. It is straight- forward that convergence is reached in one step if we replace the scalar relaxation pa- rameterα by the diagonal matrix = diag(1/λ1, . . . ,1/λN). This situation is quite caricatural, but it suggests looking for generalizations of the scheme with a matricial relaxation parameter instead of a scalar one and, in fact, to treat the unknowns differ- ently: this is what we call a multiparameter extension of the relaxation process. Being a natural approach, MP schemes have however recently reemerged from a dynamical point of view steeming from the theory of inertial manifolds and the nonlinear Galerkin method [19,23].
Several results have been already obtained for the solution of stationary problems:
in [5,12] multiparameter extensions of nonlinear Richardson iterative processes have been successfully proposed for the solution of bifurcation problems and in [3] multipa- rameter generalisations of Lanczos methods were developped. The aim of this article is to show that this approach can also be used for the numerical time integration of evolu- tion equations with a special focus on the finite differences case. We associate iterative relaxation processes to temporal integration procedures in order to build efficient MP schemes with the aid of the scales decomposition provided by incremental unknowns.
More specifically, the algebraic similarity of Richardson’s relaxation and Forward-Euler integration is used to propose a modification that increases the convergence of the first and the stability of the second. In this way, an explicit and robust MP scheme is built for the time integration of evolution equations.
The present article is organized as follows: first, in section 2, we briefly present how multiparameter extensions of both nonlinear and linear descent schemes have been obtained. After that, in section 3, we recall some techniques for generating several structures from a given function. In section 4 we consider time evolutive problems and we present MP generalizations of classical one-step methods and in section 5, we show through the solution of CFD problems that these new schemes, which are difficult to obtain with a classical approach, have an important improved stability.
2. Multiparameter methods 2.1. MP Richardson methods
Let us start with the linear problem
Findu∈Rn such that
Au=b, (1)
whereAis ann×n positive definite matrix and b is a given vector inRn. We solve problem (1) numerically by looking to the family of Richardson schemes
Letu0be given and set
uk+1=uk−αk(Auk −b), k1, (2) whereαk is a real positive parameter.
In (2), all the components ofrk = Auk −b are treated in the same manner, say, relaxed with the same parameterαk. However, as mentioned in the introduction,ukmay contain blocks of components with distinct behaviors, representing different physical quantities and, as we will see in section 3, this situation explicitly arises when uk is written using decomposition methods such as the incremental unknowns.
Since the fundamental idea of multiparameters methods is to relax each component ofukwith a different relaxation parameter in order to obtain faster convergence, we write the iterative schemes in (2) as:
Letu0be given and set
uk+1=uk −k(A uk−b), k1, (3) wherek is the diagonal matrix
k =
αk,1In1 0 . . . 0 0 0 αk,2In2 . . . 0 0 ... ... ... ... 0 0 . . . 0 αk,mInm
,
αk,i, i = 1, . . . , m, are real parameters and Ini’s the ni ×ni identity matrices, i = 1, . . . , m.
The new scheme (3) is a multiparameter extension of (2). When the αk,i do not depend onk, we obtain an extension of a Richardson method with a constant descent matrix. This approach was proposed in [12] for solving nonlinear eigenvalues problems when the blocks correspond to incremental unknowns levels, see section 3. Generaliza- tions of the Marder–Weitzner scheme were obtained, see also [7, and references therein].
Clearly, improved convergence rates are obtained if we compute at each iteration the matrixk such that the residualrk (or, in the nonlinear case, an approximation of it) is minimized in a suitable norm. However, this is a rather difficult problem to solve, since we have to chooseαk,i,i = 1, . . . , m, minimizing(I −Ak)rk. In [5] it was
proposed to express (3) in a equivalent form, introducing a rectangular matrixn×m Zk
and the vectorλk =(α1,k, . . . , αm,k)Tsuch that
Zkλk =krk. (4)
Thus, the optimal vectorλk can be computed by: the residualrk =Auk−bsatisfies the recurrence relation
rk+1 =rk−AZkλk.
Hence, the vectorλk which minimizes the euclidian norm ofrk+1is given by λk=
(AZk)TAZk
−1
(AZk)Trk, and from the consistence condition (4) we have
Zk =
(rk)1
0 . .. 0
(rk)m
. (5)
Similar extensions are proposed in [5] for the numerical solution of a bifurcation prob- lem.
2.2. MP descent methods
The approach presented above can be used for extending a large class of descent methods such as the Lanczos method. The general scheme of a descent method for solving (1) consists in computinguk+1fromukby
uk+1=uk+λkzk. (6) Hereλk is the descent steplength andzk is the search direction vector. The multipara- meter extension of (6) was proposed in [3] and consist, as in the steepest descent, in replacing the real parameterλkby a relaxation matrixk. Proceeding as above, we have
uk+1=uk+Zkk. (7)
When one considers the conjugate gradient method,kis defined byZkTk =ZkTAZkrk
and the scheme becomes
J (uk+1)=J (uk)−1 2
ZkTrk, ZkTAZk
−1
rk
, (8)
whereJ (u)= 12(Au, u)−(b, u)is the functional to be minimized. One can find in [3]
(to which we refer the reader for more details) the definition and the properties of this family of methods for different definitions of the sequenceZk leading then to multipara- meter extensions of conjugate gradient and bi-conjugate gradient methods, see [10] for an application to the solution of elliptic PDEs.
Remark 1. Other works on multiparameter extensions of methods in numerical linear algebra have been realized; let us mention the work of Molina and Raydan (2000) on GMRES [21]. MP extensions of Uzawa and augmented Lagrangian methods were also proposed, see [10].
3. Large and small scales decomposition
Let us denote byY the large scales and byZ the small ones. One of the goals of the nonlinear Galerkin methods, which were proposed for the long time integration of dissipative problems, is to modelize an enslavement of the small wavelengthes by the large ones through an exact or an approximate law (see [22,23]). In this article, instead of taking this approach, we will use the different behaviors of theY and Z quantities, which have been established and numerically observed in many situations [17,18,20] in order to propose a MP scheme that treats differently each group of scales.
When considering spectral discretizations, the separation ofY andZis straightfor- ward,
U = 2n
i=1
αiwi = n
i=1
αiwi + 2n i=n+1
αiwi =Y +Z,
since the distinct scales appear naturally due to the convergence of the series. Here the numbersαi denote real or complex coefficients and (ωi)i∈N is a spectral basis, as the Chebyshev polynomials or the Fourier basis (see [15]). When finite differences or finite elements are used, the above decomposition is not possible, since all the unknowns have the same order of magnitude, namely, that of the considered function in the physical space. Below, we present the construction of the distinct scales in the finite differences case through the incremental unknowns method [23], which consists in generating the several structures through hierarchical preconditioners; hierarchical basis are use in the finite element case (see [1] for a survey) allowing the generation of different sets of structures [20].
Let! be a domain in Rn, n = 1,2, with space step h = 1/(2N ). We denote this grid byGh, the fine grid, and byGH the coarse grid with stepH = 1/N. In one dimension, the gridsGhandGh\GH are as in the diagram below, where the coarse grid is composed of the points with even indexes, marked with×.
. o × o × o × o × o .
In 2D, the fine gridGh are the points(ih, j h),i, j =1, . . . ,2N −1, and the points of the coarse grid are of the form(2ih,2j h):
. . . . . o o o o o o o . . o × o × o × o . . o o o o o o o . . o × o × o × o . . o o o o o o o . . o × o × o × o . . o o o o o o o . . . . . .
The first step for the construction of the IU is to separate the nodal unknowns according to the grid they belong to. The variables corresponding to the coarse grid are denoted by Y and the ones related to Gh \GH are named Uf. The incremental unknownsZ are defined by the change of variable
Z =Uf −R◦Y, (9)
whereR:GH → Gh\GH is apth order interpolation operator. The incremental un- knowns Z are small in magnitude and according to the Taylor’s formula, their size is O(hp)(forp=2, this fact has been demonstrated, forn=1,2,3, see [13,14]).
The discretization in more levels is accomplished by recursively defining theZi’s through
y uf1
uf2
... ufd
=S
y z1
z2
... zd
, (10)
where due to (9),Shas a lower triangular form. In this case, thanks to the properties of compression of the data, one can organize the components of each vector intomblocks of componentsvi, i = 1, . . . , mof respective sizeni, m
i=1ni = nthat havea priori different orders of magnitude.
For instance, let us explicitly state the discrete equations defining the IU in the case ofp=2 [14,23]. In dimension one, letUj,j =0, . . . ,2N −1, be the nodal unknowns onGh, we set
Z2j+1=U2j+1−1
2(U2j+U2j+2), j =0, . . . , N −1, (11)
U0=U2N =0, (12)
which means that the IUs are defined as increments to the values ofU at the fine grid points of the average of the values ofUat the surrounding coarse grid points. This same
definition applies to the 2D case, however we have to consider 3 different situations, as shown below
×◦
× × ◦ × × ×
× ◦ ×. (13)
Thus, the incremental unknowns are defined by Z2i,2j+1=U2i,2j+1−1
2(U2i,2j+U2i,2j+2), Z2i+1,2j=U2i+1,2j−1
2(U2i,2j+U2¯ı+2,2j), Z2i+1,2j+1=U2i+1,2j+1− 1
4(U2i,2j+U2i+2,2j+U2i,2j+2+U2i+2,2j+2), fori, j =0, . . . , N−1 andUα,β =0 ifαorβ ∈ {0,1}.
4. Dynamical system approach to multiparameter methods
We now use the separation of scales provided by the IU discretization and go fur- ther on looking at the iterative processes as temporal integration procedures in order to build efficient extensions of multiparameter schemes by adding insights steeming from the nonlinear Galerkin theory. Given the linear system
AX=F, (14)
one can search for solutions from a dynamical system point of view and solve the asso- ciated PDE
Xt+AX =F (15)
looking for steady-states. In fact, Richardson’s iteration applied to (14) amounts to the iterative process
rn+1=rn−αArn, (16)
for the residualrn = AXn−F, which is equivalent to the Forward-Euler integration method applied to (15) withαplaying the role of the time step parameter. This equiva- lence implies the same stability restrictionα∈ ]0,2/ρ(A)[for both methods.
From the time integration point of view, this time step restriction is naturally related to the need of capturing the faster temporal evolution of the structures associated to the largest eigenvalues of the operatorA. Thus, if we could somehow slow down these faster modes, a bigger value ofα would be allowed and faster convergence of Richardson’s iteration to the steady-states would be insured by the extra stability of the Forward-Euler method.
Stepping ahead on this direction, we apply the variable transformation
W =eβρ(A)tX, (17)
to (15) and scheme (16) to the resulting equation Wt +
A−βρ(A)I
W =eβρ(A)tF,
where the parameter β is positive, and we can clearly see that it can be adjusted to provide an operator with a smaller spectral radius, allowing the use of a bigger time step. At the end of each iteration, we apply the inverse transformation
Xn+1=e−βρ(A)tWn+1, (18)
resulting in the scheme
Xn+1=e−1tβρ(A)
I−1tA+1tβρ(A)I
Xn+1tF
, (19)
where we considert =0 at the begining of each iteration andα=1t.
The variable transformation (17) was proposed in [16] in order to improve the temporal stability of explicit methods applied to the solution of a parabolic equation when considering an spectral discretization in space. It indeed provides extra stability to the iterative process, however it changes the steady-state solutions of (15). Note that due to the time discretization the transformation (17) appears as a first order polynomial modification in (19); on the other hand, the inverse transformation (18) is expressed as a (much stronger) exponential correction. The solution is to approximate the inverse transformation also as a first order rational correction as below:
Xn+1=e−1tβρ(A)Wn+1≈
I+1tβρ(A)−1
Wn+1, (20) obtaining the scheme
Xn+1=Xn− 1t 1+1tβρ(A)
AXn−F
. (21)
The above scheme keeps unaltered the steady-state solutions of (15) (see [16]) and it allows the use of bigger values for1t, depending on the parameterβ. The new stability restriction is given by
1t < 2
ρ(A)(1−2β), (22)
yielding unconditional stability forβ = 12. Note that scheme (21) can be rewritten as
Xn+1−Xn+1tβρ(A)
Xn+1−Xn
+1tAXn=1tF,
which after division by1tcan be seen as a discretization of the time dependent equation Xt +γ βρ(A)Xt +AX =F,
with γ = 1t. While yelding faster convergence for stationary problems, the above scheme loses accuracy when applied to time-dependent problems. This comes from the fact that the extra stability is obtained through a modification of equation (15) that leads to a temporal inconsistency. Since the size of the error depends on the temporal variation
ofX, we use the IU discretization of section 4 and only apply (21) to the small scales.
Thus, by using the usual notations, after the introduction of d levels of incremental unknowns for the numerical solution of (16), we obtain the scheme:
yn+1 zn1+1 ... znd+1
=
yn zn1 ... zd
−1tG
S−1AS
yn zn1 ... zd
−S−1F
, (23)
with
G=
Id 0 . . . 0
0 1
1+1tβ1ρ(A1)Id 0
... . ..
0 0 1
1+1tβdρ(Ad)Id
,
whereAi is the restriction of the operatorAto the subspace generated by the scalezi. The algebraic similarity with 3 shows that the above scheme is nothing else but a multiparameter scheme.
The above result has been improved in [16] for the case of a spectral discretization to obtain a temporal consistent modification with increased stability. However, since our goal in this work is to show that the conjugation of MP schemes and scales separation can be of use for time dependent problems, scheme (23) will do for our purposes.
Remark 2. Results concerning a detailed error analysis for the above scheme can be found in [16] for the spectral discretization case and in [11] for the IU. For time depen- dent problems with steady state solutions, equation (21) provides a robust, yet explicit, time integration scheme (see the numerical results on section 4). The size of the error introduced is in straight connection with the size of the temporal variation ofX, which makes these types of schemes suitable for conjugation with temporal adaptivity.
5. Numerical results
In the numerical experiments below, we compare the classical Forward-Euler scheme with the MP scheme (23) for the numerical solution of the 2-D Burgers equation on the domain!= ]0,1[2:
ut− 1
Re1u+uux+vuy =f1 in!, (24)
vt − 1
Re1v+uvx+vvy=f2 in!, (25)
with the following initial and forcing functions:
U0(x, y)=cos(π x)cos(πy), V0(x, y)= x+y 4 ,
f1(x, y, t)=sin(3π x)sin(4πy)+3 sin(16π x)sin(8πy),
f2(x, y, t)=sin(4π x)sin(2πy)+3 sin(10π x)sin(16πy), Re =100, which provide very rich dynamics to the solution before it attains its steady-state (see [6]). Since the velocity components u and v have not the same behavior, we use distinct family of parametersβ for their time integration, leading to the following dis- cretization:
Un+1=Un−1tG1
1
ReAUn+N L1
Un, Vn
−f1
, (26)
Vn+1=Vn−1tG2
1
ReAVn+N L2
Un, Vn
−f2
, (27)
where we have set Gi =diag
1, 1
1+γ1(i)1t, 1
1+γ2(i)1t, . . . , 1 1+γm(i)1t
and γj(i)=βj(i)ρ(A).
Adenotes the discretization matrix of−1on the grid andN L1(respectivelyN L2) the discretization of the nonlinear term inU (respectively inV). Also, notice that one can chose different numbers of incremental unknowns levels forU and forV.
Four levels of IU were used in the MP scheme and the values of the parametersβ were determined by the improved stability condition (22) in order to allow a time step 3 times larger than the maximum one for the classical scheme to each block of scales.
Since, higher wavenumbers imply in bigger spectral radii for the operatorA, the values ofβ increase with the level index of the IU decomposition. The values below were empirically chosen to be
β1(1) =80, β2(1) =200, β3(1)=400, β4(1)=1000, β1(2) =20, β2(2) =100, β3(2)=300, β4(2)=600.
Both solutions were computed up tot =4.0, with1t =10−3for the classical scheme (maximum time step allowed is 1.6×10−3) and1t = 5×10−3 for the MP scheme.
Both schemes had 127 points in each direction.
In figure 1, we see that the dynamical behavior of both solutions is similar, how- ever, due to the larger time step, CPU time for the MP scheme is 4 times smaller than the classical scheme. In figure 2, we show the numerical profiles obtained (no visual difference), with the discreteL2norm of the difference between the two solutions being 4.87×10−3 for the U-component and 3.94×10−3 for the V-component. Moreover, note that the MP scheme has successfully captured the high gradients, showing that no harm was done to the relevant structures of the numerical solution.
We have here presented results on 2d Burgers equations and restricted to the second order IUs on uniform meshes but similar performances are obtained when considering other type of IUs (higher order, on nonuniform meshes), see [10].
(a)
(b)
(c)
Figure 1. 2-D Burger’s equation. (a) Evolution of the Euclidian norm of the discrete derivative ofuvs. time for classical and MP Euler methods. (b) Evolution of the Euclidian norm of the discrete derivative ofvvs.
time for classical and MP Euler methods. (c) Time vs. CPU time for classical and MP Euler’s methods.
(a) (b)
Figure 2. 2-D Burgers equation. Computed solution (velocity field) att = 4.0. (a) MP Euler’s method.
(b) Classical Euler method.
6. Conclusion and perspectives
In this article we showed that the generalization of the relaxation step together with the use of a tool of separation of scales allows to extend classical schemes in directions that are otherwise difficult to reach. The multiparameter approach presented is versatile and applies also to other types of spatial discretization. In a general way, MP extensions of iterative processes show that the ideas contained in the nonlinear Galerkin methods, say treating in a different way blocks of unknowns that behave differently, apply to more general situations that those coming from the long time integration of dynamical systems: several fields of numerical analysis are concerned as, e.g., convergence accel- eration [4], numerical linear algebra [3,21] and numerical methods for PDEs [5,7,12], for stationary problems.
Many developements of the MP approach are possible, in particular block (or grid) strategies and construction of adaptive multiparameter schemes for the long time inte- gration of dissipative dynamical systems are features to be considered; time adaptive schemes will be presented in a forthcoming work.
Acknowledgements
The second author was partly supported by CNPq grant 300-315/98-8 and by FAPERJ grant E-26/170.216/2000. Both authors thank Roger Temam for the fruitful discussions.
References
[1] R.E. Bank, Hierarchical bases and the finite elements method, in:Acta Numerica, 1996 (Cambridge Univ. Press, Cambridge, 1996) pp. 1–43.
[2] C. Brezinski, Variations on Richardson’s method and acceleration, in:Numerical Analysis. A Numer- ical Analysis Conf. in Honour of Jean Meinguet, Bull. Soc. Math. Belg. (1996) 33–44.
[3] C. Brezinski, Multiparameter descent methods, Linear Algebra Appl. 296 (1999) 113–142.
[4] C. Brezinski, Acceleration procedures for matrix iterative methods, Note Ano 423 (2001).
[5] C. Brezinski and J.-P. Chehab, Multiparameter iterative schemes for the solution of systems of linear and nonlinear equations, SIAM J. Sci. Comput. 20(6) (1999) 2140–2159.
[6] C. Calgaro, J. Laminie and R. Temam, Dynamical multilevel schemes for the solution of evolution equations by hierarchical finite element discretizations, Appl. Numer. Math. 23 (1997) 403–442.
[7] J.-P. Chehab, A nonlinear adaptive multiresolution method in finite differences with incremental un- knowns, Math. Modelling Numer. Anal. 29 (1995) 451–475.
[8] J.-P. Chehab, Incremental unknowns method and compact schemes, Math. Modelling Numer. Anal.
32(1) (1998) 51–83.
[9] J.-P. Chehab and A. Miranville, Incremental unknowns method on nonuniform meshes, Math. Mod- elling Numer. Anal. 32(5) (1998) 539–577.
[10] J.-P. Chehab and B. Costa, Multiparameter extensions of iterative processes, Rapport Technique du Laboratoire de Mathématiques d’Orsay, RT-02 (2002).
[11] J.-P. Chehab and B. Costa, Time explicit schemes for spatial finite differences, J. Sci. Comput. (2003) to appear.
[12] J.-P. Chehab and R. Temam, Incremental unknowns for solving nonlinear eigenvalue problems: New multiresolution methods, Numer. Methods Partial Differential Equations 11 (1995) 199–228.
[13] M. Chen, A. Miranville and R. Temam, Incremental unknowns in finite differences in three space dimensions, Comput. Appl. Math. 14(3) (1995) 219–252.
[14] M. Chen and R. Temam, Incremental unknowns for solving partial differential equations, Numer.
Math. 59 (1991) 255–271.
[15] B. Costa and L. Dettori, Fourier collocation splittings for partial differential equations, J. Comput.
Phys. 20 (1999) 469–475.
[16] B. Costa, L. Dettori, D. Gottlieb and R. Temam, Time marching techniques for the nonlinear Galerkin method, SIAM J. Sci. Comput. 23(1) (2001) 46–65.
[17] A. Debussche, T. Dubois and R. Temam, The nonlinear Galerkin method: A multiscale method ap- plied to the simulation of homogeneous turbulent flows, Theoret. Comput. Fluid Dyn. 7(4) (1995) 279–315.
[18] O. Goubet, Nonlinear Galerkin methods using almost-orthogonal finite element bases, Nonlinear Anal. Theory Methods Appl. 20(3) (1993) 223–247.
[19] M. Marion and R. Temam, Nonlinear Galerkin methods, SIAM J. Numer. Anal. 26 (1989) 1139–1157.
[20] M. Marion and R. Temam, Nonlinear Galerkin methods; The finite element case, Numer. Math. 57 (1990) 205–226.
[21] B. Molina and M. Raydan, Spectral variants of Krylov subspace methods, Numer. Algorithms 29 (2002) 1–3, 197–208.
[22] R. Temam,Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. (Springer, Berlin, 1997).
[23] R. Temam, Inertial manifolds and multigrid methods, SIAM J. Math. Anal. 21 (1990) 154–178.
