A numerical approach of Friedrichs systems under constraints in bounded domains

Transcription

1 A numerical aroach of Friedrichs systems under constraints in bounded domains Clément Mifsud, Bruno Desrés To cite this version: Clément Mifsud, Bruno Desrés. A numerical aroach of Friedrichs systems under constraints in bounded domains <hal > HAL Id: hal htts://hal.archives-ouvertes.fr/hal Submitted on 16 Nov 016 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 A numerical aroach of Friedrichs systems under constraints in bounded domains Clément Mifsud and Bruno Desrés Abstract We resent here an exlicit finite volume scheme on unstructured meshes adated to first order hyerbolic systems under constraints in bounded domains. This scheme is based on the work [3] in the unconstrained case and the slitting strategy of [4]. We show that this scheme is stable under a Courant-Friedrichs-Lewy condition (and convergent for roblems osed in the whole sace) and we illustrate the solution constructed by this scheme on the examle of the simlified model of erfect lasticity. From the theoretical oint of view, the interaction between the constraint and the boundary of the domain in the model of erfect lasticity is encoded by a nonlinear boundary condition. With this numerical aroach, we will show that, even if this scheme uses the underlying linear boundary condition, the results are consistent with the nonlinear model (and in articular with the nonlinear boundary condition). Mathematics Subject Classification 010: 65M08, 65M1, 35L50, 35L60, 74C05 1 Introduction The aim of this article is to examine the numerical aroximation of Friedrichs equations under constraints (osed in the whole sace or in bounded domains). To do so, we use a oular method for hyerbolic roblems: the method of finite volumes (for a detailed resentation of this method, we refer to [5, 6]). Although there is an imortant number of schemes that have been develoed, the analysis of the convergence and its rate of schemes on unstructured meshes for multidimensional Clément Mifsud Université Pierre et Marie Curie Paris 6, CNRS, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005, France, mifsud@ljll.math.umc.fr Bruno Desrés Université Pierre et Marie Curie Paris 6, CNRS, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005, France and Institut Universitaire de France 016, desres@ann.jussieu.fr 1

3 Clément Mifsud and Bruno Desrés roblems (i.e. the domain is a subset of R n with n > 1 and the solution belongs to R m with m > 1) is still in its infancy. However, the article [9] has established a rate of convergence for the RDG scheme (see []), using P0 finite elements in sace and the R1 scheme in time, on unstructured meshes for generic Friedrichs systems of the following form { t U + n j=1 j (A i U) + BU = f, in (0,T ) R n, U(0,x) = U 0 (x), in R n (1), where U : (t,x) (0,T ) R n R m, A i : (t,x) (0,T ) R n M m m sym, B : (t,x) (0,T ) R n M m m, f : (t,x) (0,T ) R n R m and M m m (res. M m m sym ) is the sace of m m (res. symmetric) matrices with real coefficients. A similar analysis has been erformed in the note [3] on bounded domains. In addition, the study of the convergence of a scheme based on the Rusanov scheme on Cartesian meshes has been erformed in [4] for constrained Friedrichs systems. In fact, to show the existence of a weak solution (in the sense of Definition 1) to the constrained Friedrichs system { t U + n j=1 A j j U = 0 in (0,T ] R n ; U(0,x) = U 0 (x) if x R n, U(t,x) C if (t,x) [0,T ] R n (), where C is a fixed closed and convex subset of R m (with 0 C ), the authors construct a numerical solution with a two ste scheme such that a subsequence converges to a weak solution of (). In this aer, we extend the strategy of [4] to schemes on unstructured meshes and to roblems osed in bounded domains. In Section, we recall some notations and define our finite volume scheme on unstructured meshes for constrained Friedrichs systems in bounded domains. In Section 3, we recall some results of [4] on constrained Friedrichs systems in the whole sace and state a convergence result in the whole sace on a similar scheme (to the one resented in Section on bounded domains). This result tells us that the finite volume scheme on unstructured meshes, based on the work [9], associated with a rojection ste has the same rate of convergence (in the sace L ((0,T ) R n ;R m )) as in the unconstrained case (obtained in [9]). In Section 4, we show that the scheme resented in Section is stable (under a Courant-Friedrichs-Lewy condition) in the sace L (0,T ;L (Ω,R m )). Then in Section 5, we briefly recall the equations of the simlified model of the dynamical erfect lasticity roblem (described in [1]) and how this roblem is related to the constrained Friedrichs systems. Finally, in Section 6, we illustrate the solution constructed by this scheme on the examle of the simlified model of the dynamical erfect lasticity roblem and show that the interaction between the constraint and the boundary condition that has been underlined theoretically by the nonlinear boundary condition can also be observed numerically.

4 A numerical aroach of Friedrichs systems under constraints in bounded domains 3 Descrition of the scheme In this section, we resent the general framework of this work and the scheme we are interested in. Let T h be a triangulation of Ω R n (a n-dimensional olytoe) i.e. T h = ( i ) i I, with I N, is a family of oen nonemty convex olytoe such that i I i = Ω, for all i j, i j = /0 and h = su i I (diam i ) < +. The set of edges of a olytoe is denoted E. We introduce the following notations (see also Figure 1), m,m : L n -measure of,h n 1 -measure, e E : an edge ((n 1)-dimensional olytoe) of with H n 1 -measure m e, E i,e b : the set of interior edges e of,the set of boundary edges e of, ν e : the unit exterior normal of on the edge e with ν e =(ν 1 e,ν e,...,ν n e ), e : neighboring cell of with e = e. e ν e e Fig. 1 An unstructured meshes of the square [0,1] [0,1]. Here the olytoes are triangles. We also suose that the triangulation is regular in the sense that there exists a constant C 1 > 0 (indeendent of the triangulation T h ) such that T h, C 1 h n m, and T h, e E C 1 h n 1 m e. We want to investigate the numerical aroximation (using finite volume schemes) of the following constrained Friedrichs system { t U + n i=1 A i i U = f, on (0,T ) Ω; U(0,x) = U 0 (x), on Ω, (3) (A ν M ν )U = 0, on (0,T ) Ω; U(t,x) C, a.e in (0,T ) Ω where C R m is a closed convex (indeendent of t and x) with 0 C, A ν = n i=1 A iν i with ν = ( ν 1,...,ν n) is the unit exterior normal to Ω and M ν is a nonnegative symmetric matrix that encodes the boundary condition and has to satisfy some algebraic conditions (see [8, Section.1]). Remark 1. In articular, due to the hyotheses on A ν and M ν, we have

5 4 Clément Mifsud and Bruno Desrés 1. For all k R m, there exists a unique trile (k 0,k,k + ) such that k = k 0 +k +k + and k 0 kera ν, k (ker(a ν M ν )) ImA ν and k + (ker(a ν + M ν )) ImA ν.. For all k,κ R m, k A ν κ = k A ν κ + k + A ν κ +. The equations of (3) have to be understood in a weak sense (see Definition 1 for the case Ω = R n and Section 5 for the general case). To aroximate the solutions of this kind of roblem, we first forget about the constraint and use a finite volume scheme (exlicit in time) based on the note [3]. More recisely, we use a iecewise constant aroximation of U, denoted by V h, such that (t,x) [t,t +1 ), V h (t,x) = v, with v0 = 1 U 0 (x)dx, m where 0 = t 0 < t 1 < < t N+1 = T (t +1 t = t) and in a first ste, we construct m ( t v +1, v ) + e E g e m e = f := 1 m t t +1 t f (t,x)dxdt, where A e = n i=1 A iν i e and we define the interior fluxes (e Ω = /0), g e = (A e ) + v + (A e ) v }{{} e, (4) }{{} Outcoming flow from to e Incoming flow in from e where we denote (A e ) (res. (A e ) + ) the negative (res. ositive) art of A e, and the (centered) boundary fluxes, g e = A e + M e v, (5) with M e = M νe a matrix satisfying the conditions of [8, Section.1] (see also Remark 1). In order to take account of the constraint, we simlify roject on each cell the value v +1, onto the set C. Hence, the second ste is ( ) v +1 = P C v +1,. where P C is the rojection onto C. It leads us to the following scheme for U 0 L (R n ;C ), T h, v 0 = m 1 U 0(x)dx, T h, 0 N, v +1, = v m t e E g e m e + t f ( ), T h, 0 N, v +1 = P C v +1,. (6) Thanks to the following discrete Green formula e E A e m e = 0 e E b A e m e + e E i (A e ) + m e = e E i (A e ) m e, (7)

6 A numerical aroach of Friedrichs systems under constraints in bounded domains 5 one can rewrite the first ste of the scheme (6) in a non conservative form v +1, v m e = t (A e ) (v e E m m e v e ) i e E b M e A e v m + f. (8) Remark. We denote by ; the canonical scalar roduct of R m and. the associated norm. By abuse of notation, we also use the notation. for the (matrix) oerator norm associated with the canonical norm of R m. Remark 3. When Ω = R n, one can use the scheme (6) to aroximate the solution of the roblem (). In that case, all the sums over E b are emty sums. 3 Previous results on constrained Friedrichs systems in the whole sace The aim of this section is to recall the definition of weak solutions to Friedrichs systems under convex constraints in the whole sace and to state some numerical results about these systems. We consider the following Cauchy roblem: find U : [0,T ] R n R m such that { t U + n j=1 A j j U = 0 in (0,T ] R n ; U(0,x) = U 0 (x) if x R n, U(t,x) C if (t,x) [0,T ] R n (9), where C is a fixed (i.e. indeendent of the time and sace variables) non emty closed and convex subset of R m containing 0 in its interior, the matrices A j are m m symmetric matrices indeendent of time and sace, and T > 0. This tye of nonlinear hyerbolic roblems has been introduced in [4] where a notion of weak solutions to roblem (9) has been defined. Definition 1. Let U 0 L (R n,c ) and T > 0. A function U L ([0,T ] R n,c ) is a weak constrained solution of (9) if we have for all κ C and φ Cc ([0,T [ R n ) with φ 0, T Rn ( 0 U κ t φ + n j=1 U κ;aj (U κ) j φ) dxdt We recall here the existence and uniqueness result of [4]. + U 0 (x) κ φ(0,x)dx 0. R n (10) Theorem 1. Assume that U 0 L (R n,c ). There exists a unique weak constrained solution U L ([0,T ] R n,c ) to (9) in the sense of Definition 1. The existence of a solution has been obtained in [4] thanks to a finite volume scheme on Cartesian grids. At each time ste, the scheme first let the solution evolve

7 6 Clément Mifsud and Bruno Desrés according to the Rusanov scheme without taking care about the constraint. Then, on each mesh they roject the solution onto the set of constraints. Thanks to this slitting strategy and to a comactness argument (which relies on the fact that the mesh is Cartesian), they show that the numerical solution admits a convergent subsequence and they rove that the limit of this subsequence has to be a solution of (9) in the sense of Definition 1. In this aer, we use this slitting strategy for schemes defined on unstructured meshes. One can show that the scheme (6) (see Remark 3) enjoys the same rate of convergence as in the unconstrained case (for the comlete roof, see [7]). Theorem. Let U H 1 ((0,T ) R n ;C ) be a dissiative solution associated with the initial condition U 0 H 1 (R n ;C ). Let V h be the solution constructed from U 0 thanks to the scheme (6) (see Remark 3). Then we have, U V h L ((0,T ) R n ;R m ) C h, for some constant C deending on ε, n, T, U 0 and the matrices A i. 4 Stability in time of schemes Once we know that the strategy of [4] combined with the scheme, analyzed in [9], leads to a convergent scheme (on unstructured meshes) for constrained Friedrichs systems in (0,T ) R n, one can analyze this slitting strategy on bounded domains (i.e. for Problem (3)). In this section, we rove that the scheme (6) enjoys a stability roerty under a Courant-Friedrichs-Lewy condition. For simlicity, we decide to derive this stability roerty in the case where the source term is null. In that case, the L (R n )-norm of the solution do not increase in time. Proosition 1. Suose that the following CFL condition holds: ( max tm su (Ae ) tm, su (M e A e )/,e E m,e E b m ) 1, (11) the scheme (6) is stable, i.e. the aroximate solution V h satisfies (here f 0) t [0,T ], V h (t, ) L (R n ;R m ) U 0 L (R n ;R m ). Proof. From the non-conservative form (8), we have where we set v +1, = e E m e v +1, (e), m

8 A numerical aroach of Friedrichs systems under constraints in bounded domains 7 { v + tm v +1, (e) = v tm m m (A e ) (v v e ), if e E i, M e A e v, if e E b. Observe that we have for all e E i, since (A e ) M m m sym, v, (e) = v tm m + tm m v v e ; ( v ;(A e ) v ( + v e ;(A e ) v e ) Id + tm m (A e ) )(A e ) (v v e ) Using the CFL condition, we obtain that ( y R m, Id + tm ) (A e ) y;y 0. (1) m In articular, if we aly (1) to y = ( (A e ) ) 1/ (v v e ), it yields v, (e) v tm ( v m ;(A e ) v + v e ;(A e ) v ) e. (13) Now, if e E b, we have, again since A e and M e belong to M m m sym, v +1, (e) = v tm v m ; M e A e tm ( Me A e Id tm m m v ( Me A e Similarly, the CFL condition (11) imlies that for all y R m, we have Id tm ( ) Me A e y;y 0, m and algebraic maniulations (see Remark 1) tells us that )) v ;v. (14) ( Me A e Id tm ( )) Me A e v m ;v ( = Id tm ( )) Me A e M 1/ m e (v ) +;M 1/ e (v ) + 0, which imlies that (14) becomes Using convexity, it yields v +1, (e) v tm m v ; M e A e v.

9 8 Clément Mifsud and Bruno Desrés v +1, v t m ( v ;(A e ) v + v e ;(A e ) v ) me e e E i t m Furthermore, if we use the relation (7), we obtain v +1, v t m e E b v ; M e A e v ( v ;(A e ) + v + v e ;(A e ) v ) me e e E i t m e E b v ; A e + M e v Remark that, thanks to Remark 1, we have for all e E b v ; A e + M e v = (v ) ;M e (v ) 0. Consequently, from (15) and since for all y R m, P C (y) y, we obtain m e. m e. (15) v +1 v t ( m v ;(A e ) + v + v e ;(A e ) v ) me e. (16) e E i Then, we remark T h ( v ;(A e ) + v + v e ;(A e ) v ) me e = 0. e E i Consequently, summing the inequality (16) over T h and from = 0 to q 1, where t [0,T ] and q an integer such that t [t q,t q+1 ) (or q = N +1 if t = T ), leads to the stability roerty. 5 The simlified model of the dynamical erfect lasticity Let us briefly recall the equations of this model and the two oints of views that one can use to describe its (theoretical) solution. First, the equations, derived from the hysics of solids (see [1, Sections 3.1 & 3.]), of this simlified model of dynamical erfect lasticity are { t v divσ = f, v = t σ + t, (17) σ 1, and σ; t = t. where v : Ω [0,T ] R is the velocity of the material, σ : Ω [0,T ] R the Cauchy stress tensor and : Ω [0,T ] R the lastic deformation tensor and Ω

10 A numerical aroach of Friedrichs systems under constraints in bounded domains 9 is a oen bounded subset of R. The tensor σ is constrained to stay in the unit closed Euclidean ball of R, denoted B. To these equations, we add initial and boundary conditions. The boundary condition, that comes from the hyerbolic oint of view, is the following nonlinear one σ;ν + T (v) = 0, on (0,T ) Ω, (18) where T (z) = min( 1,max(z,1)). It shows a threshold on the velocity (due to the constraint) in the boundary condition. We also need an initial condition that has to satisfy two hyotheses (v,σ)(t = 0) = (v 0,σ 0 ) (19) σ 0 ;ν + v 0 = 0 H 1 on Ω, (0) σ 0 1 a.e. in Ω. (1) The first condition asserts that the initial condition has to satisfy the hyerbolic boundary condition that one could use in the unconstrained case and the second condition states that the initial condition satisfy the constraint. In fact, one can show (see [1, Proosition 7.1]) that the solution of this simlified model satisfies the following inequality for all (k,τ) R B and all ϕ W 1, (R R ) (with ϕ 0 and comactly suorted in R R ) T ( (v k) + σ τ ) t ϕ dxdt + ( (v0 k) + σ 0 τ ) ϕ(0)dx 0 Ω T 0 Ω Ω (σ τ) ϕ(v k)dxdt + T 0 Ω f (v k)ϕ dxdt T + (σ ν τ ν)(t (v) k)ϕ dh n 1 dt 0. () 0 Ω Thanks to (18) and algebraic maniulations, one has (σ ν τ ν)(t (v) k) = 1 ( (k + τ ν) (T (v) k (σ ν τ ν)) ) (k + τ ν), (3) Equation (3) allows us to rewrite (), using the hyerbolic variable U = t (v,σ) as T 0 Ω + U κ t ϕ + Ω i=1 U 0 κ ϕ(t = 0)dx + U κ;a i (U κ) i ϕ + F;U κ ϕ dxdt T 0 Ω where F = t ( f,0,0), U 0 = t (v 0,σ 0 ), κ = t (k,τ) κ + ;M ν κ + ϕ dh n 1 (x)dt 0, (4)

11 10 Clément Mifsud and Bruno Desrés A 1 = , A = and M ν = 0 (ν 1 ) ν 1 ν, (5) ν 1 ν (ν ) and κ + stands for the rojection onto (ker(a ν + M ν )) ImA ν. The fact that Equation (4) is satisfied for all κ and all ϕ is the definition of a solution to Problem (3) (see also [8]). In addition, when the solution U is in W 1, ([0,T ];L (Ω;C ), one can show (see [1, Section 7]) that Equations (17), (18) and (19) are equivalent to this definition of a weak constrained solution to Problem (3). 6 Numerical tests on the simlified model of the dynamical erfect lasticity Now that this mechanical roblem has been ut into the hyerbolic framework (3), the simlified model of dynamical erfect lasticity can be aroached thanks to the scheme described in Section. One imortant oint to notice first is that this scheme does not include a secial treatment at the boundary to model the nonlinear boundary condition (18). Indeed, we only take into account the constraint thanks to a rojection ste on every mesh and the first ste of this scheme uses the linear boundary condition (A ν M ν )U = 0 σ;ν + v = 0. (6) Our goal now is to test numerically the interactions between the boundary condition and the constraint for this articular hyerbolic system under constraint and to see if the nonlinear boundary condition is obtained with this scheme. The major oint that allows us to bring to light these facts is the velocity threshold overrun in the boundary condition (18). To observe this overrun, we resent here one test case (for more test cases, see [7, Section 4.4]). The test is based on the following formal motivation: we want to observe large velocities near the boundary. But if we look at the equation of motion t v divσ = f, we see that if f is ositive (for examle) near the boundary (for each time) then the velocity is going to increase over time near the boundary. Hence, we resent a test case when the source term f is equal to a ositive constant near the boundary and to zero elsewhere. This test allows us to obtain large velocity near the boundary (i.e. v 1 near Ω) and to bring to light that the nonlinear boundary is taken into account by our scheme. For this test case, we use the following data Satial domain : Ω = [0,1] [0,1]. Our mesh is regular and contains triangles.

12 A numerical aroach of Friedrichs systems under constraints in bounded domains 11 Final time : T = 1. We use 800 time-stes and consequently the CFL condition (11) is aroximately equal to Initial data : In this test, we use data that touch the boundary x = 1. The initial velocity v 0 is null outside the oen ball B 1 of radius 0.3 and center (1,0.5), v 0 is equal to 1 on the oen ball B of radius 0.5 and center (1,0.5). In the stri between these two balls, we join these two constants using a C 1 connection. It is imortant to notice that 1 v 0 0. In order to satisfy the (linear) boundary condition at x = 1 the first comonent of σ is equal to v 0. The second comonent of σ is null on Ω. Consequently, we have v 0 + σ;ν = 0 on Ω. Remark also that the initial data belong to the convex set of constraints. The term source f is equal to 50 for all t [0,T ], for all y [0,1] and x > 0.8 and to 0 elsewhere. We decide to highlight the interaction between the constraint and the boundary at time t = 0.5 in Figure. In this figure, we dislay the velocity (to left of the figure), the first comonent, denoted σ 1 in the following, of σ (to right), the second comonent (bottom left), denoted σ, and the term σ 1 + T (v) (which is involved in the boundary condition at x = 1: σ 1 + T (v) = 0). We observe that the introduction of a ositive term source in the stri [0.8,1] [0,1] allows us to get a large velocity (i.e. v 1) near the boundary x = 1 (see Figure a). The theoretical boundary condition imlies that in this situation we should see that σ 1 = 1 at the boundary x = 1 (and consequently, σ = 0 due to the constraint). Numerically, the scheme roduces a solution that matches the mathematical model (see Figure b and c). Consequently, the nonlinear boundary condition is satisfied by the numerical aroximation (see Figure d) desite the fact that we have not imlemented any articular treatment at the boundary to get this nonlinear boundary condition. This fact may be seen as a first validation of our scheme. References 1. Jean-François Babadjian and Clément Mifsud. Hyerbolic structure for a simlified model of dynamical erfect lasticity. arxiv: Bernardo Cockburn and Chi-Wang Shu. TVB Runge-utta local rojection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Com., 5(186): , Yves Coudière, Jean-Paul Vila, and Philie Villedieu. Convergence d un schéma volumes finis exlicite en tems our les systèmes hyerboliques linéaires symétriques en domaines bornés. C. R. Acad. Sci. Paris Sér. I Math., 331(1):95 100, Bruno Desrés, Frédéric Lagoutière, and Nicolas Seguin. Weak solutions to Friedrichs systems with convex constraints. Nonlinearity, 4(11): , Robert Eymard, Thierry Gallouët, and Rahaèle Herbin. Finite volume methods. In Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, ages North-Holland, Amsterdam, Randall J. LeVeque. Finite volume methods for hyerbolic roblems. Cambridge Texts in Alied Mathematics. Cambridge University Press, Cambridge, Clément Mifsud. Variational and hyerbolic methods alied to constrained mechanical systems. PhD thesis, Université Pierre et Marie Curie, 016.

13 1 Clément Mifsud and Bruno Desrés (a) Velocity (b) First comonent σ 1 of σ (c) Second comonent σ of σ (d) σ 1 + T (v) (i.e. the boundary term on the right of the domain) Fig. : Test case at time t = Clément Mifsud, Bruno Desrés, and Nicolas Seguin. Dissiative formulation of initial boundary value roblems for Friedrichs systems. Comm. Partial Differential Equations, 41(1):51 78, Jean-Paul Vila and Philie Villedieu. Convergence of an exlicit finite volume scheme for first order symmetric systems. Numer. Math., 94(3):573 60, 003.