A method of Lagrange Galerkin of second order in time. Une méthode de Lagrange Galerkin d ordre deux en temps

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1 A metod of Lagrange Galerkin of second order in time Une métode de Lagrange Galerkin d ordre deux en temps Jocelyn Étienne a a DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, Great-Britain. Tel.: , Fax : Abstract Te Lagrange Galerkin metod is te coupling of a finite element metod for space discretisation wit te metod of caracteristics for te discretisation of te material derivative in some parabolic problems. We propose a new sceme of second-order accuracy in time, wic in contrast wit previous metods does not require a correction term. Numerical examples, including Burgers equation, illustrate te convergence rate and low computational cost of te metod. To cite tis article: J. Étienne, C. R. Acad. Sci. Paris, Ser. I 336 (003). Résumé La métode de Lagrange Galerkin consiste à coupler une discrétisation aux éléments finis en espace avec la métode des caractéristiques pour la discrétisation de la dérivée matérielle dans certains problèmes paraboliques. Nous proposons un nouveau scéma, d ordre deux en temps, ne faisant pas intervenir de terme de correction comme les métodes précédentes. Des exemples numériques, dont l équation de Burgers, illustrent le taux de convergence et l efficacité en temps de calcul de la métode. Pour citer cet article : J. Étienne, C. R. Acad. Sci. Paris, Ser. I 336 (003).. Introduction Te metod of caracteristics consists in using te caracteritic curves of an advection field a in order to rewrite te transport operator ( t + a ) in terms of a partial derivative in time. Indeed, if one introduces te caracteristics X defined for any (x, s) Ω [0, T] by: dx {x, s; t} = a(x{x, s; t}, t) dt for t [0, T], () X{x, s; s} = x address: J.Etienne@damtp.cam.ac.uk (Jocelyn Étienne). Preprint submitted to Elsevier Science 5t October 005

2 one notes tat by te cain rule, u(x{x, s; t}, t) t = [( ) ] t + a u (X{x, s; t}, t) () If te advection field is interpreted as te velocity of a continuous medium, tis transform is te passage from a Eulerian frame to te Lagrangian frame. Tis metod was employed in te context of finite elements by Bercovier and Pironneau [] as a particular upwinding metod, were te left and side derivative of () is discretised by a backward Euler finite difference and te instant s is cosen as te current time step t n, so tat X{, s; t n } is te identity. For a parabolic problem of te type ( ) t + a u + Au = f in Ω, u(, t 0 ) = u 0, u Ω [0,T] = 0, a time-step by te metod of caracteristics is: u n u n X{, t n ; t n } + Au n = f(, t n ), (3) and allows to calculate an approximation u n of u(, t n ) if te caracteristic mapping X{, t n ; t n } and an approximation u n of u(, t n ) are known, wit a first order accuracy (see e.g. []). Te transport problem in Eq. (3) is idden in te term u n X{, t n ; t n }. Te calculation of tis term cannot be considered separately from te discretisation in space employed, and tus we now introduce our finite elements discretisation. First we write a variational problem to solve, Given u n V, find u n V suc tat (un, v) + a(u n, v) = f(, t n ) + ( u n X{, t n ; t n }, v ) v V (4) were V is an appropriate subspace of H0(Ω), and a(v, v ) = (Av, v ) for all v, v V. Let V V be a finite element space of degree k based on a triangulation T, equiped wit a projection operator Π suc tat for f V, s = 0 or, f Π f s,ω C k s+ f k+,ω Ten te discrete problem is, Given u n V, find u n V suc tat (un, v ) + a(u n, v ) = Π f(, t n ) + ( ) u n X{, t n ; t n }, v v V (5) were Π is a projection operator from V onto V. Te advected field transfer, tat is, te calculation of te term ( u n ) X{, t n ; t n }, v is not standard, because in general u n X{, t n ; t n } V, and can be done in two main different ways [3], tat we will briefly describe. In one approac, u n X{, t n ; t n } is treated as a function analytically known at every point of te domain, and quadrature formulae are used eiter directly, or after interpolating u n X{, t n ; t n } by a function in V. Alternatively, advantage can be taken of te fact tat u n belongs to V by calculating a triangulation T = (T X {, t n ; t n }) T, were X {, t n ; t n } is an approximation of X{, t n ; t n }, and te projection ũ V can be calculated suc tat: (ũ, v ) = v u n (X {, t n ; t n }) dx v V. (6) K T K because bot v and u n (X {, t n ; t n }) are polynomial on K T. Te caracteristic feet X{, t n ; t n } (for te first approac of te advected field transfer) or X{, t n ; t n } (for te second) are not known exactly in general, and need to be numerically integrated from Problem (). For a first order sceme, a Euler meod is enoug to obtain te desired accuracy, wit:

3 X{x, t n ; t n } = x + a(x, t n ) + O( ), X{x, t n ; t n } = x a(x, t n ) + O( ). (7) Te space discretisation as to be done in a finite element space of degree larger tan k. Tus, if k, isoparametric elements elements of degree k are needed if one wants to calculate T. In te case of nonlinear transport (a = u), te approximation u n of a(, t n) is not known by te time wen we need to approximate X{, t n ; t n }, but one notes tat using u n instead does not affect te order of accuracy in Eq. (7). Tis will not be te case anymore wen considering a second order sceme. It is ten easy to ceck formally tat (un ũ ) is a discretisation of ( t + a ) u to te first order in time, and it is possible to prove an error estimate in k+s + as is done in [] for te incompressible Navier Stokes equations. We will now consider te extension of tis sceme to second-order time accuracy.. Second order scemes Te difficulty of extending te Lagrange Galerkin sceme to second-order time accuracy was pointed out in [4]. Indeed, a time-sceme suc as: u n u n X{, t n ; t n } + (Aun + Au n ) = (f(, t n ) + f(, t n )) (8) is not second order accurate, because we ave: t + a u(x{, t n ; t n / }, t n / ) = (un u n X{, t n ; t n }) + O( ), but te point X{, t n ; t n / } is at a distance of order of x, and we ave only: t + a u(x, t n / ) = (un u n X{, t n ; t n }) + O(). Tus a correct second order sceme writes, u n u n X{, t n ; t n } + (Aun X{, t n ; t n } + Au n ) = (f(x{, t n; t n }, t n ) + f(, t n )) were Au n X{, t n ; t n } is understood as (Au n ) X{, t n ; t n }. Te problem is tat, if one wants to apply te Green formula to te variational form of Eq. (9), te cange of variable X{, t n ; t n } produces an additional correction term. If A is te Laplace operator, tis term is ( J n u n X{, t n ; t n }, v ), were J n is te Jacobian matrix of a(, t n ). Altoug calculations are possible wit tis discretisation [4], we sow in tis Note tat a two-step time discretisation allows to write a simpler sceme, wic as te advantage to be independent of te form of operator A, and can be naturally extended to te nonlinear transport case a = u. Let us consider te two-step sceme, u u n = Au n + f(, t n ), (0a) u n u X{, t n ; t n } + Au n = f(, t n ). (0b) No Au n X{, t n ; t n } appears in Eqs. (0a) and (0b), so te Green formula can be applied as usual to teir variational form, witout te Jacobian of te transform X{, t n ; t n } appearing. On te oter and, if we compose te first equation by X{, t n ; t n } and sum te two, we obtain exactly Eq. (9), wic implies tat u n constructed in tis way is a second-order accurate approximation of u(, t n ). 3 (9)

4 (a) Figure. Error max n u n u(, t n) 0 (b) versus for (a), problem () and (b), problem (). +, algoritm (3), and 0 - -, algoritm (0). Te triangles ave a ratio of and, respectively. Note tat te first step of tis sceme is only te explicit calculation of some auxiliary variable u, and( is computationally very ceap. Tis auxiliary variable can be interpreted as u = u(, t n ) + t + a ) u(, t n ) + O(). Tis is interesting, because an approximation of appropriate order of te mapping X{, t n ; t n } writes: X{x, t n ; t n } = x + a(x, t n ) + t + a u(, t n ) + O( 3 ), and tus in te nonlinear transport case a = u, we can calculate te carateristic mapping very easily wit: X{x, t n ; t n } = x + u + O( 3 ). 3. Numerical examples We present numerical examples for one-dimensional test problems and compare te result to teir analytical solution. Te finite element space V cosen is piecewise linear (k = ) on a 000-elements regular mes and te advected field transfer is done by projection. Te first problem is linear, wit te rigt-and-side f cosen suc tat te exact solution is u(x, t) = x( x)( + cost) V : u (x, t) + x( x)sin(t) u t x (x, t) + u (x, t) = f(x, t) (x, t) (0, ) [0, ] () x u(x, 0) = x( x), u(0, t) = u(, t) = 0 Te convergence of algoritms (3) and (0) is sown in Fig. (a). Tis is te case only for te advected field transfer by projection, for te quadrature metod, it is necessary to obtain a second-order approximation X {x, tn; t n } of X{x, t n; t n }, for instance wit Eq. (7), and ten to calculate X{x, t n; t n } = x + u X {x, tn; t n } + O( 3 ). 4

5 Te second problem is known as Burger s viscous equation, wic involves a nonlinear transport term, and tis time te rigt-and-side f is suc tat te exact solution is u(x, t) = x( x)t: u (x, t) + u(x, t) u t x (x, t) + u (x, t) = f(x, t) (x, t) (0, ) [0, ] () x wit te same initial and boundary conditions. Te convergence of algoritms (3) and (0) is sown in Fig. (b). For te same timestep, te CPU time is found to be only around 30% more wit te second order accurate metod. Tis overcost sould be even lower in dimensions iger tan one, as te resolution of te linear system in te implicit step increases faster tan te one of oter operations. 4. Perspectives Tis novel metod of second-order in time for Lagrange Galerkin discretisation of parabolic problems does not require to introduce a correction term but only te explicit calculation of an auxilliary variable. Te case of nonlinear transport is treated, and a metod making use of te same auxilliary variable is introduced to approximate te caracteristic curves in tat case. Numerical results confirm te order of convergence of te metod. Tis metod can be directly applied to muc more complicated problems, suc as te Navier Stokes equations. It can also be used for free-boundary flow problems, since te application of tis tecnique in te context of te Arbitrary Lagrangian Eulerian metod (ALE) is staigtforward. A detailed analysis of te second-order ALE metod will be presented in a fortcoming paper [5]. References [] M. Bercovier and O. Pironneau. Caracteristics and te finite elements metod. In Tadaiko Kawai, editor, Finite element flow analysis, proceedings of te 4t Int. Symp. on Finite Elements Metods in Flow Problems, Tokyo, pages Nort-Holland, 6 9 July 98. [] E. Süli. Convergence and nonlinear stability of te Lagrange-Galerkin metod for te Navier-Stokes equation. Numer. Mat., 53: , 988. [3] A. Priestley. Exact projections and te Lagrange-Galerkin metod: A realistic alternative to quadrature. J. Comput. Pys., :36 333, 993. [4] H. Rui and M. Tabata. A second order caracteristic finite element sceme for convection-diffusion problems. Numer. Mat., 9:6 77, 00. [5] J. Étienne. Te second order Lagrange Galerkin metod for free surface flows. in preparation. 5