The present work is devoted to the proof of uniqueness of the solution of the finite elements scheme in the case of variable coefficients. Finite elements method is applied for the numerical solution of the mixed problem for symmetric hyperbolic systems with variable coefficients. Moreover, dissipative boundary conditions and its stability are proved. Finally, numerical example is provided for the two dimensional mixed problem in simply connected region on the regular lattice. Coding is done by DELPHI7

Davlatov, Sh. O.

Djuraevich, Aloev Raxmatillo

Eshkuvatov, Zainidin K.

Nik Long, Nik Mohd Asri

Universiti Putra Malaysia Institutional Repository

Malaysian Journal of Mathematical Sciences 10(S) August: 49–60 (2016)Special Issue: The 7th International Conference on Research and Education in Mathematics(ICREM7)MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCESJournal homepage: http://einspem.upm.edu.my/journalUniqueness Solution of the Finite ElementsScheme for Symmetric Hyperbolic Systems withVariable CoefficientsR.D. Aloev ∗1, Sh.O. Davlatov 2, Z. K. Eshkuvatov 3, andN.M.A. Nik Long 41Faculty of Mathematics and Mechanics, National University ofUzbekistan, Tashkent, Uzbekistan.2Faculty of Physics and Mathematics, Karshi State University,Uzbekistan.3Faculty of Science and Technology, Universiti Sains IslamMalaysia (USIM), Negeri Sembilan, Malaysia and Institute forMathematical Research (INSPEM), Universiti Putra Malaysia(UPM), Selangor, Malaysia.4Department of Mathematics, Faculty of Science, Universiti PutraMalaysia (UPM), Selangor, Malaysia and Institute forMathematical Research, Universiti Putra Malaysia (UPM),Selangor, Malaysia.E-mail:aloevr@mail.ru∗Corresponding authorABSTRACTThe present work is devoted to the proof of uniqueness of the solution ofthe finite elements scheme in the case of variable coefficients. Finite ele-ments method is applied for the numerical solution of the mixed problemfor symmetric hyperbolic systems with variable coefficients. Moreover,dissipative boundary conditions and its stability are proved. Finally,numerical example is provided for the two dimensional mixed problemin simply connected region on the regular lattice. Coding is done byDELPHI7.R.D. Aloev, Sh.O. Davlatov, Z. K. Eshkuvatov, and N.M.A. Nik Long1. The Statement of the Problem ([2],[3])There are many results [(D.Aloev et al., 2014),(Falk and Richter, 1999),(Duet al., 1988),(Hughes and Tezduyar, 1984),(Segerlind, 1976),(Adjerid and Bac-couch, 2010) and (Codina, 2008)] that are devoted to the study of finite elementmethod (FEM) for a mixed problem for symmetric hyperbolic systems with thehelp of finite differences method. Various explicit and implicit finite differenceschemes (FDS) are obtained. The stability of the obtained schemes is also in-vestigated. Error estimates of the approximate solutions of the mixed problemsare given. Particularly, in the preceding work of the authors [1], the stabilityof the implicit difference scheme obtained by the finite elements method fortwo dimensional t - hyperbolic system with constant coefficient were proved.A detailed review of [(Falk and Richter, 1999),(Du et al., 1988),(Hughes andTezduyar, 1984),(Segerlind, 1976),(Adjerid and Baccouch, 2010) and (Codina,2008)] is given in [(D.Aloev et al., 2014)].We consider the following t - hyperbolic system of the formA(t, x, y)∂u∂t+B(t, x, y)∂u∂x+ C(t, x, y)∂u∂y+D(t, x, y)u = F (t, x, y), (1)in the domainG ={(t, x, y) : t ∈ (0, T ), (x, y) ∈ Ω},whereΩ = {(x, y) : 0 < x < lx, |y| <∞}with the boundary conditions:For x = 0, 0 < t ≤ T, |y| <∞ :uI(t, 0, y) = S(x, y)uII(t, 0, y) + g1(x, y), (2)for x = lx, 0 < t ≤ T, |y| <∞ :uII(t, lx, y) = R(x, y)uI(t, lx, y) + g2(x, y), (3)for |y| → ∞ and 0 < t ≤ T, 0 ≤ x ≤ lx :‖u‖ → 0, ‖u‖ =√√√√ N∑k=1u2k (4)and with the initial datau(0, x, y) = u0(x, y), (x, y) ∈ Ω. (5)50 Malaysian Journal of Mathematical SciencesUniqueness Solution of the Finite Elements Scheme for Symmetric Hyperbolic Systems withVariable CoefficientsHere {t, x, y} are independent variables andB(t, x, y) = (bks(t, x, y)), C(t, x, y) = (cks(t, x, y)),are symmetric matrices of N×N dimension, the elements of these matrices aregiven functions, and A is the positive defined matrix; D(t, x, y) = (dks(t, x, y))is the square matrix of N ×N dimension and R, S are rectangle matrices, inwhich the number of rows are respectively equals to the number of the positiveand negative eigenvalues of the matrix B.In equations (1)-(5), the functionu(t, x, y) = (u1, u2, · · · , uN )Tis unknown vector function andF (t, x, y) = (f1, f2, · · · , fN )Tis the given function.Approximation of the domain G and the description of the finite elements meth-ods is given in [(D.Aloev et al., 2014)]. High accuracy finite element methodsare given in [(Hughes and Tezduyar, 1984),(Segerlind, 1976),(Adjerid and Bac-couch, 2010) and (Codina, 2008)] for different type of hyperbolic equations.The present work is devoted to the proof of uniqueness of the solutionof the finite elements scheme in the case of variable coefficients. Finite ele-ments method is applied for the numerical solution of the mixed problem forsymmetric hyperbolic systems with variable coefficients. Moreover, dissipativeboundary conditions and its stability are proved. Finally, numerical exampleis provided for the two dimensional mixed problem in simply connected regionon the regular lattice. Coding is done by DELPHI7.The paper is arranged as follows: In Section 2, construction of the differencescheme for symmetric t-hyperbolic system is described. Section 3 discusses theuniqueness of the solution of the finite elements scheme. Finally, numericalexample is provided in Section 4 to show the accuracy and efficiency of themethod. Section 5 concludes the main ideas of the the approximate method.Malaysian Journal of Mathematical Sciences 51R.D. Aloev, Sh.O. Davlatov, Z. K. Eshkuvatov, and N.M.A. Nik Long2. Construction of the Difference Scheme forSymmetric t-Hyperbolic SystemWe divide the domain Ω to the finite elements which have not generic inte-rior points. The finite element is denoted by K. Then it is clear thatΩ =⋃K⊂ΩK.Next, the interval [0, T ] is divided into Nt parts and the partitioning points aretk = τ · n, (n = 0, ..., Nt), τ = TNt.In each time layer tn the approximate solution uh (tn, x, y) of the mixed problem(1)-(5) is searched in the formunh = uh (tn, x, y) =∑(xi;yj)∈ΩunijQij(x, y),where Qij(x, y) is the base function [5], its values at the nodes (xi, yj) ∈ Ω areequal to one, and in other points are equal to zero. Letunij = u(xi, yj , tn) =u1ij(tn)u2ij(tn)...uNij(tn) =un1ijun2ij...unNijOn the element K with the nodes Mij we approximate equation (1) by theimplicit difference scheme:(A(tn+1, x, y)un+1h − unhτ,Qij)K+(B(tn+1, x, y)∂un+1h∂x,Qij)K+(C(tn+1, x, y)∂un+1h∂y,Qij)K+(D(tn+1, x, y)un+1h , Qij)K= (F (tn+1, x, y), Qij)K , (xi, yj) ∈ Ω, (6)where (u, v)K =∫∫Ku(x, y) · v(x, y)dK.52 Malaysian Journal of Mathematical SciencesUniqueness Solution of the Finite Elements Scheme for Symmetric Hyperbolic Systems withVariable CoefficientsLet us rewrite the difference scheme (6) for (xi, yj) ∈ Ω in the following form(A(tn+1, x, y)un+1h , Qij)K+ τ(B(tn+1, x, y)∂un+1h∂x,Qij)K+ τ(C(tn+1, x, y)∂un+1h∂y,Qij)K+ τ(D(tn+1, x, y)un+1h , Qij)K= τ (F (tn+1, x, y), Qij)K + (A(tn+1, x, y)unh, Qij)K . (7)3. Uniqueness of the Solution of the FiniteElements SchemeFor simplicity we suppose that A is unit matrix. Assume that the mixedproblem (1)-(5) has a unique solution and the boundary conditions satisfy thefollowing conditions [4]: ∫Γ(Ω)Su · uds ≥ 0 ∀t ∈ [0, T ] . (8)D +D∗ − ∂B∂x− ∂C∂y≥ 0, (9)where S = nxB + nyC and n = (nx, ny) is the unit external normal to the Ω,and Γ(Ω) is the boundary of the domain Ω.Consider the following differential-difference system:Lu ≡ τB(t, x, y)∂u∂x(t, x, y) + τC(t, x, y)∂u∂y(t, x, y)+ (I + τ ·D(t, x, y))u(t, x, y)= u(t− τ, x, y) + τ · F (t, x, y). (10)Here I is the unit matrix, t is the time divisibly by τ .We consider the bilinear forma(u, v) ≡ (Lu, v)K , (11)where(u, v)K =∫Ku · vdK. (12)Malaysian Journal of Mathematical Sciences 53R.D. Aloev, Sh.O. Davlatov, Z. K. Eshkuvatov, and N.M.A. Nik LongLet Pm(K) be the set of polynomials of order ≤ m defined onK, the coefficientsdo not depend on t. Then for each element of K the following equality holdsa(uh, vh) = (uh(t− τ, x) + τ · F (t, x), vh)K , (13)where ∀vh ∈ Pm(K).If the solution is known on the layer t−τ and vh ∈ Pm(K) is the base function,then from (13) and definition of the solution on the layer t we get the systemof algebraic equations (7). Let us introduce the following operator:L∗v ≡ v − τB(t, x, y)∂v∂x(t, x, y)− τC(t, x, y)∂v∂y(t, x, y) + τ(D∗(t, x, y)− ∂B∂x(t, x, y)− ∂C∂y(t, x, y))v(t, x, y). (14)Lemma 1. The following equality holdsa(u, v) = (u, L∗v)K + τ∫Γ(K)Su · v (15)Using integration by parts we obtaina(u, v) ≡ (Lu, v)K=(τB∂u∂x+ τC∂u∂y+ (I + τ ·D)u, v)K= τ ∫Γ(K)Su · v+ ((I + τ ·D∗)v, u)K−(B∂v∂x+ C∂v∂y, u)K−((∂B∂x+∂C∂y)v, u)K=(v − τB ∂v∂x− τC ∂v∂y+ τ(D∗ − ∂B∂x− ∂C∂y)v, u)K+ τ∫Γ(K)Su · v.Lemma is proved.Lemma 2. The following inequality is truea(u, u) ≥ (u, u)K + τ2∫Γ(K)Su · u. (16)54 Malaysian Journal of Mathematical SciencesUniqueness Solution of the Finite Elements Scheme for Symmetric Hyperbolic Systems withVariable CoefficientsDoing some transformations we get the following chain of inequalities:(u, L∗u)K =(u− τ(B∂u∂x+ C∂u∂x)+ τ(D∗ − ∂B∂x− ∂C∂y)u, u)K= −(u+ τ(B∂u∂x+ C∂u∂y)+ τDu, u)K+(2u+ τDu+ τ(D∗ − ∂B∂x− ∂C∂y)u, u)K= −a(u, u) +((2I + τ(D +D∗ − ∂B∂x− ∂C∂y))u, u)K(17)Taking into account (9), using equalities (15) and (17) we get the followinginequality:a(u, u) =12((2I + τ(D +D∗ − ∂B∂x− ∂C∂y))u, u)K+τ2∫Γ(K)Su · u ≥ (u, u)K +τ2∫Γ(K)Su · u. (18)Theorem 1. If the solution of the finite elements scheme of type uh ∈ Pn(K)exists and is convergent, then it is uniquely defined on K and satisfy the fol-lowing inequality:‖uh(t, x, y)‖2Ω ≤ eT ‖uh(0, x, y)‖2Ω + (T + 1)(eT − 1)F, (19)where ‖u‖Ω =√∫Ωu · u, F = maxt∈[0,T ]‖Fh(t, x, y)‖2Ω .Assume that the values of the convergent solution uk−1h ∈ Pm(K) on thelayer tk−1 = τ(k− 1) are defined and the base function vh ∈ Pm(K) is chosen,then for definition of the values of the convergent solution ukh ∈ Pm(K) on thelayer tk = τk from equality (11) we get the following system of linear algebraicequations :τ(B(tk, x, y)∂uh∂x(tk, x, y) + C(tk, x, y)∂uh∂y(tk, x, y))K+((I + τ ·D(tk, x, y))uh(tk, x, y), vh)K= (uh(tk−1, x, y) + τ · Fh(tk, x, y), vh)K . (20)We rewrite the system (20) in the matrix form:Ahuh = bh (21)Malaysian Journal of Mathematical Sciences 55R.D. Aloev, Sh.O. Davlatov, Z. K. Eshkuvatov, and N.M.A. Nik LongIf we prove that the homogenous system of algebraic equations Ahuh = 0has only trivial solution uh ≡ 0, then the uniqueness of the solution of thenonhomogeneous system (21) holds. This problem is equivalent to the proof ofthe equality uh = 0 on K, i.e.τ(B(tk, x, y)∂uh∂x(tk, x, y) + C(tk, x, y)∂uh∂y(tkx, y))K+((I + τ ·D(tk, x, y))uh(tk, x, y), vh)K= 0, ∀vh ∈ Pn(K). (22)We can take vh = uh, then taking into account the fact that the left part ofequality (20) is a(uh, vh). From Lemma 3.2 we will get the following inequality:(uh, uh)K +τ2∫Γ(K)−Γ∗(K)Shuh · uhds+ τ2∫Γ∗(K)Shuh · uhds ≤ 0, (23)where Γ∗(K) = Γ(K) ∩ Γ(Ω).Because of arbitrariness of the element K ∈ Ωh from (23) we obtain the fol-lowing inequality:(uh, uh)Ω +τ2∑K∈Ω∫Γ(K)−Γ∗(K)Shuh · uhds+ τ2∫Γ(Ω)Shuh · uhds ≤ 0. (24)It is obvious that ∑K∈Ω∫Γ(K)−Γ∗(K)Shuh · uhds ≡ 0. (25)Taking of equalities (25) from (24) we get the following:(uh, uh)Ω ≤ 0. (26)From the last inequality it follows that uh ≡ 0. According to Lemma 3.2 thefollowing inequality is true:2(unh, unh)K + τ∫Γ(K)−Γ∗(K)Shunh · unhds+ τ∫Γ∗(K)Shunh · unhds≤ 2(un−1h + τ · Fnh , unh)K , (27)where ukh = uh(tk, x, y), uk−1h = uh(tk−1, x, y) and Fkh = Fh(tk, x, y). From (27)56 Malaysian Journal of Mathematical SciencesUniqueness Solution of the Finite Elements Scheme for Symmetric Hyperbolic Systems withVariable Coefficientsit follows that2|unh|2K + τ∫Γ(K)−Γ∗(K)Shunh · unhds+ τ∫Γ∗(K)Shunh · unhds≤ 2(un−1h + τ · Fnh , unh)K≤ ∥∥un−1h + τ · Fnh ∥∥2K + ‖unh‖2K . (28)From the inequality (28) we obtain|unh|2K + τ∫Γ(K)−Γ∗(K)Shunh · unhds+ τ∫Γ∗(K)Shunh · unhds≤ ∥∥un−1h + τ · Fnh ∥∥2K≤ (∥∥un−1h ∥∥K + τ‖Fnh ‖K)2≤ (τ + 1) ∥∥un−1h ∥∥2K + (τ2 + τ) ‖Fnh ‖2K . (29)Taking into account the identity (25) and the condition (8), from inequality(29) it follows that|ukh|2Ω ≤ (τ + 1)∥∥uk−1h ∥∥2Ω + (τ2 + τ)∥∥F kh∥∥2Ω . (30)From the last inequality we have:|uh(t, x, y)|2Ω ≤ eT ‖uh(0, x, y)‖2Ω+ (T + 1)(eT − 1)F, (31)where F = maxt∈[0,T ]‖Fh(t, x, y))‖2Ω. Theorem is proven.4. Numerical ComputationsFor the numerical solution of the mixed problem (1)-(5) the complex pro-grammes Delphi-7 is applied. For the solution of the obtained system of alge-braic equations "The method of minimal discrepancy" is used. Example 1:As model problem in the domainΩ = {(x, y) : 0 < x < 2, 0 < y < 2}we consider the following system∂u1∂t + y∂u1∂x + x∂u1∂y − u2 = x2 + y2+x+ y + 2t− t2 − 4∂u2∂t − y ∂u2∂x − x∂u2∂y + u1 = t2 + xy + x+ y + 2tMalaysian Journal of Mathematical Sciences 57R.D. Aloev, Sh.O. Davlatov, Z. K. Eshkuvatov, and N.M.A. Nik Longwith the boundary conditionsfor x = 0: u1 = t2;for x = 2: u2 = 2− y + t2;for y = 0: u2 = 2− x+ t2and with the initial data for t = 0:{u1 = xy,u2 = 4− x− y.The exact solution of this mixed problem is as follows:u1 = xy + t2; u2 = 4− x− y + t2.It is not difficult to check that above mentioned mixed problem satisfies con-ditions of Theorem 3.1. The values of the error is computed‖u− v‖ , ‖u‖Ω =√√√√∫Ωu · uTable 1 given below show the error values of the numerical solution for thevalues of the parameters Nx = 10, 20; Ny = 10, 20; t = 10. Here v is thenumerical solution of the mixed problem by the finite elements methodTable 1: Error values of Example 1Nt Nx = 10, Ny = 10 Nx = 20, Ny = 2010 5.2580955 2.986575920 2.6233082 1.495197640 1.3014823 0.747724980 0.6395239 0.3731776160 0.3107552 0.1865482Table 1 shows that the error in the numerical solution tends to zero whenthe difference grid steps tend to zero. Since the error of approximation is offirst order O(τ, hx, hy), this result is acceptable. Pay attention to the range ofintegration (at the Nx = Ny = 20, a step value is hx = hy = 0.1 and henceO(τ, hx, hy) = O(0.1)).58 Malaysian Journal of Mathematical SciencesUniqueness Solution of the Finite Elements Scheme for Symmetric Hyperbolic Systems withVariable CoefficientsExample 2: In the domainΩ = {(x, y) : 0 < x < 2, 0 < y < 2}we consider the following system∂u1∂t + y∂u1∂x + t∂u1∂y + yu1 − xu2 = xt(y − t)+t2 − x3 + (x+ y + y2)(1 + t),∂u2∂t − t∂u2∂x + x∂u2∂y + xu1 − yu2 = 2(t+ xy)+xt(x+ y − 2)− y3 + yt2with the boundary conditionsfor x = 0: u1 = yt;for x = 2: u2 = 4 + y2 + t2;for y = 0: u1 = xt; u2 = x2 + t2and with the initial data for t = 0:{u1 = xy,u2 = x2 + y2.The exact solution of this mixed problem is as follows:u(x, y, t) =(u1u2)=(xy + xt+ ytx2 + y2 + t2)It is not difficult to check that above mentioned mixed problem satisfies condi-tions of Theorem 3.1. The error values of the numerical solution for the valuesof the parametersnt = 20, nx = 20, ny = 20in ‖u‖ = √J (t) is equal‖u− v‖ = 0.1201078.Here v is the numerical solution of the mixed problem by the finite elementsmethod.5. ConclusionIn this paper, finite element method is used to find approximate solution ofsymmetric hyperbolic systems with variable coefficients. Stability of the finiteelement method is proved. Numerical examples indicates that the proposedmethod is highly efficient for the tested problem.Malaysian Journal of Mathematical Sciences 59R.D. Aloev, Sh.O. Davlatov, Z. K. Eshkuvatov, and N.M.A. Nik LongAcknowledgmentThis work was supported by University Putra Malaysia under Geran UPMGP-i/2014/9442300ReferencesAdjerid, S. and Baccouch, M. (2010). Asymptotically exact a posteriori errorestimates for a one-dimensional linear hyperbolic problem. Appl. Numer.Math., 60:903–914.Codina, R. (2008). Finite element approximation of the hyperbolic wave equa-tion inmixed form. Comput. Methods Appl. Mech. Engrg., 197:1305–1322.D.Aloev, R., Eshkuvatov, Z., Davlatov, S., and Long, N. N. (2014). Sufficientcondition of stability of finite element method for symmetric t-hyperbolicsystems with constant coefficients. Computers and mathematics with appli-cations, 68:1194–1204.Du, Q., Gunzburger, M., and Layton, W. (1988). A low dispersion, high accu-racy finite element method for first order hyperbolic systems in several spacevariables. Comput. Math. Appl., 15(68):447–457.Falk, R. and Richter, G. (1999). Explicit finite element methods for symmetrichyperbolic equations. SIAM J. Numer. Anal., 36(3):935–952.Hughes, T. and Tezduyar, T. (1984). Finite element methods for first-orderhyperbolic systems with particular emphasis on the compressible euler equa-tions. Comput. Methods Appl. Mech. Engrg, 45:217–284.Segerlind, L. J. (1976). Applied Finite Element Analysis. John Wiley & Sons,Inc., 289-308 n edition.60 Malaysian Journal of Mathematical Sciences

Uniqueness solution of the finite elements scheme for symmetric hyperbolic systems with variable coefficients

http://psasir.upm.edu.my/52357/1/4.%20Rakhmatilo.pdf

Uniqueness solution of the finite elements scheme for symmetric hyperbolic systems with variable coefficients

Abstract

Similar works

Full text

Available Versions

Universiti Putra Malaysia Institutional Repository