CONSTRUCTION AND RESEARCH OF ADEQUATE COMPUTATIONAL MODELS FOR QUASILINEAR HYPERBOLIC SYSTEMS

. In the paper, we study a class of three-dimensional quasilinear hyperbolic systems. For such system, we set the initial boundary value problem and construct the energy integral. We construct the diﬀerence scheme and obtain an a priori estimate for its solution.


1.
Introduction. Finite-difference methods are often used to find the numerical solutions of quasilinear hyperbolic systems. There are many different ways to form difference schemes for quasilinear hyperbolic systems, some of approaches to do it are given in [3,1,2]. In [3], an interesting approach is described, based on the possibility of writing the system of gas dynamics equations in two variants. First we explain the meaning of this approach for a differential problem.
Here A = A(U, t, x, y, z) = A * (U, x, y, z); B = B(U, t, x, y, z) = B * (U, x, y, z); C = C(U, t, x, y, z) = C * (U, x, y, z); D = D(U, t, x, y, z) = D * (U, x, y, z); Q = Q(U, t, x, y, z) are real quadratic matrices of order N , elements of which are bounded functions; A * , B * , C * , D * are the corresponding transposed matrices; F = F (t, x, y, z), U 0 (x, y, z) are given vector-functions, vanishing at infinity; U (t, x, y, z) is an unknown vector-function of dimension N ; T, l are positive real numbers. Furthermore, A > 0 is a positive defined matrix (see [3]). We note that in [3], systems of the form (1) defined in such a way are called the symmetric t-hyperbolic systems. Assume that there is a nonsingular transformation W = W (U, t, x, y, z), that reduces the initial boundary problem (1)-(4) to the form: with periodic boundary conditions: and and with the initial data for t = 0: Here A = A(W, t, x, y, z) > 0; B = B(W, t, x, y, z); C = C(W, t, x, y, z); D = D(W, t, x, y, z); Q = Q(W, t, x, y, z) are quadratic matrices of order N ; A T , B T , C T , D T are the corresponding transposed matrices. Multiplying (on the right) the both sides of the system (5) by the vector W we have Hence, taking into account the following equalities we get the following identity ∂ ∂t Here G = G(t, x, y, z) Integrating the both sides of the identity (9) over the domain Γ t1,t2 = {(t, x, y) : The last identity was obtained in the class of functions W, for which the quadratic forms (CW, W ) and (DW, W ) tend to zero at infinity. Using periodic boundary conditions and the lemma on the integral inequality (see [3], page 152), we have the following inequality Here the constant M estimates the norm of the matrix 1 2 (Q + Q T ) and the constant N estimates the right side of equation (5).
3. On symmetrization of the gas dynamics equations. Consider the system of equations describing a three-dimensional motion of gas under the assumption that the gas is inviscid, non-heat-conducting and in the local thermodynamic equilibrium, i.e. there exists a state equation of gas. In the Cartesian coordinate system x = (x 1 , x 2 , x 3 ), the equations of gas dynamics can be written in the following conservative form where ρ is the density, ν = (υ 1 , υ 2 , υ 3 ) is the velocity, ik = ρ · υ i · υ k + p · δ ik is the momentum flow density tensor, p is the pressure, E is the inner energy, and V = 1/ρ is the specific volume. Moreover, the following thermodynamical identity holds where T is the gas temperature and S is the entropy. It follows from (12) that That is, if we add to the system (12) the state equation then we get a closed system which can be now considered as the system for finding, for example, the vector of unknowns Finally, we add to (12) the following additional conservation law (the entropy conservation) which is fulfilled on smooth solutions of the system (11). We also note that, taking into account (13), the system of equations (11) is equivalent on smooth solutions to the following non-conservative system 1 ρ · c 2 · dp dt where In [6], the Cauchy problem was studied for the system (11), and the local-in-time theorem on existence of the smooth (classical) solution was proved for this problem. S.K. Godunov [4] (see also [4,5]) proposed a special class of systems of conservation laws in the following form 1, 2, 3 are nonlinear functions of the dependent variables q i , i = 1, n (so-called productive functions). Why systems (15) are interesting? Firstly, the system (15) enables one to get an additional (n + 1)-th conservation laws. We multiply the i-th equation of (15) by q i and sum up the results: Secondly, the system (11) can be rewritten in the symmetric form where A 0 = L qiqj , A k = M k qiqj , i, j = 1, n, k = 1, 2, 3 are symmetric matrices. If, in addition A > 0 or A < 0, then (16) is a symmetric t-hyperbolic system (according to Friedrich). As a rule, we have in practice the opposite situation, when the additional conservation law is known for some equations of mathematical physics. It is the law (13), that was used in [5] (see also [6]) for constructing the symmetric system, wherea [5] s for the functions L, M k , k = 1, 2, 3, and the variables q i , i = 1, 5 (n = 5) the following expressions were obtained: Here, we consider only the case of polytrophic gas when the state equation is defined as follows (see [8]): Then, the matrices A 0 , A k , k = 1, 2, 3 are the following: We note that B 0 > 0 (if ρ > 0). With the help of simple calculations we easily find Here . By virtue of (16), the system of gas dynamics equations can be rewritten as and, in view of (17), in the form where N 1 = ( 0, 1, 0, 0, υ 1 ) * , N 2 = (0, 0, 1, 0, υ 2 ) * , N 3 = (0, 0, 0, 1, υ 3 ) * .
Integrating the equality (21) over the domain Π = {x; 0 < x k < l k , k = 1, 2, 3} and taking into account (20), we get the desired a priori estimate where While obtaining the a priori estimate (21), we also assumed that the smooth solution of system (11) has the property H(S) > 0.
We note that We now discuss the symmetrization of the gas dynamics equations (11) when the additional conservation law is which is satisfied on smooth solutions of system (11) (see, for example, [7]). Here h(S) is a smooth function of S. Simply calculating we get for the functions L, M k , k = 1, 2, 3, of the variables q i , i = 1, 5, (n = 5) the following expressions The matrices A 0 , A k , k = 1, 2, 3 take the form (the matricesB k are described above). The condition of positive definiteness of the matrixB 0 leads us to the following restrictions on the function h(S) Further, simply calculating, we find HereH · M k , k = 1, 2, 3 (the vectors M k are described above). As above the gas dynamics system (11) can be rewritten either asH or, by virtue of (25), as follows: where the vectors N k , k = 1, 2, 3 are described above. We can also (see [5], [6]) easily obtain the a priori estimatẽ While obtaining the a priori estimation (28) we assumed that the smooth solution of (11) has the propertyH (S) > 0. We note that
This gives us possibility to get energetic estimation (the difference analog a priori estimation (10)), from which it follows stability of the difference scheme (33)-(35). For this, we multiply the system (33) by the vector (1, 1, ..., 1) T : we have τ (W, W ) + r x ξ(B + W, W ) + r y η(C + W, W ) We multiply both sides of the obtained inequality by h x h y h z and sum up over i from 1 to n − 1; over j from −∞ to +∞ and over k from −∞ to +∞. Then, taking into account the following relations we obtain the following inequality which means the stability of the difference scheme (33) -(35) in the energetic norm √ J m .
As an example, we consider the hyperbolic equation (the Burgers equation) We introduce the following notations f (u) = u 2 2 , f (u) = f + (u) + f − (u), df + du ≥ 0, ∀u ∈ R, and we construct the following difference scheme for the last equation Consider the following reconstruction: U L R → R is the continuous function called limiter. ψ = 0 corresponds to the scheme of the first order, ψ = 1 is one sided scheme of the second order with upwind difference. We consider the following modification of the obtained scheme Here r = ∆/h. Now we prove that the difference model (40) admits availability of difference analogue of the energy integral. For this, we multiply the system (40) by [ U + U ]: Using formulas of difference differentiation, we have the following identity Taking into account all these transformations, we have We multiply both sides of the obtained inequality by h and sum up over i from −∞ to +∞, taking into account that the function u tends to zero at infinity and denoting the quantity h which means stability of the difference scheme in the norm √ I k . Numerical results. Consider the following problem u t + (u 2 /2) x = 0, −∞ < x < ∞, t > 0, u(x, 0) = 1, x < 0, 0, x > 0.
As parameters of numerical calculations we take −1 ≤ x ≤ 1, 0 ≤ t ≤ 1, ∆ = 0.001, h = 0.01. We rewrite the difference scheme (40) in the following form For the numerical solution of the difference scheme we apply an iterative method for nonlinear coefficients )]U i ).
In Figure 1, we give results of calculation of the numerical solution on Mathcad.
Comparing the numerical solution with the exact one, it can be concluded that the modified scheme with limiter well modulates the jump.
For clearness we consider this solution at t = 1 (see Fig. 2)