Abstract Generated abstract
This paper studies a mixed boundary value problem for a first order quasilinear hyperbolic system in a strip, with nonlocal boundary conditions involving time integrals. A finite difference grid scheme is constructed, and under Lipschitz and compatibility assumptions the paper establishes local existence of classical and generalized solutions, uniqueness of generalized solutions, and continuous dependence on perturbations of the data. It also proves convergence estimates for the grid method: first order accuracy for sufficiently regular classical solutions and order one half for generalized solutions.
Full Text
Reports of the Academy of Sciences of the USSR
1963, Volume 149, No. 3
MATHEMATICS
A. Ya. LEPIN
APPLICATION OF THE METHOD OF GRIDS TO A HYPERBOLIC SYSTEM OF QUASILINEAR EQUATIONS IN THE PLANE
(Presented by Academician S. L. Sobolev on 13 VIII 1962)
In the strip \((0 \leq t \leq t^0,\ 0 \leq x \leq x^0,\ -\infty < u_1,\ldots,u_n < \infty)\) let us consider the system of equations
\[
\frac{\partial u_1(t,x)}{\partial t}
=
\lambda_1(t,x,u)\frac{\partial u_1(t,x)}{\partial x}
+
f_1(t,x,u),
\]
\[
\frac{\partial u_2(t,x)}{\partial t}
=
\lambda_2(t,x,u)\frac{\partial u_2(t,x)}{\partial x}
+
f_2(t,x,u),
\tag{1}
\]
where \(u_1, u_2, u=\left\|\begin{matrix}u_1\\ u_2\end{matrix}\right\|\), \(f_1, f_2\) are column matrices; \(\lambda_1, \lambda_2\) are diagonal matrices, with the diagonal elements of \(\lambda_1\) nonnegative, and positive for \(x=x^0\), while the diagonal elements of \(\lambda_2\) are nonpositive, and negative for \(x=0\), with initial conditions
\[ u(0,x)=\varphi(x) \tag{2} \]
and boundary conditions
\[
u_1(t,x^0)=\alpha_1(t,u_2(t,x^0))+\int_0^t \beta_1(t,\tau,u(\tau,x^0))\,d\tau,
\]
\[
u_2(t,0)=\alpha_2(t,u_1(t,0))+\int_0^t \beta_2(t,\tau,u(\tau,0))\,d\tau.
\tag{3}
\]
Analogous systems of linear equations were studied in papers \((^{1-3})\). In studying classical and generalized solutions we shall need compatibility conditions:
\[
\varphi_1(x^0)=\alpha_1(0,\varphi_2(x^0)),
\]
\[
\varphi_2(0)=\alpha_2(0,\varphi_1(0));
\tag{4}
\]
\[ \lambda_1(0,x^0,\varphi(x^0))\varphi_1'(x^0)+f_1(0,x^0,\varphi(x^0)) = \left.\frac{\partial \alpha_1}{\partial t}\right|_{t=0,\ u_2=\varphi_2(x^0)} + \beta_1(0,0,\varphi(x^0)), \tag{5} \]
\[ \lambda_2(0,0,\varphi(0))\varphi_2'(0)+f_2(0,0,\varphi(0)) = \left.\frac{\partial \alpha_2}{\partial t}\right|_{t=0,\ u_1=\varphi_1(0)} + \beta_2(0,0,\varphi(0)). \]
To replace the corresponding boundary-value problem by a difference one, we construct a grid with step \(h\) in \(x\) and \(k\) in \(t\). Then the difference system is written as follows:
\[
u_1^{i+1\,j}
=
\varkappa \lambda_1^{ij}u_1^{i\,j+1}
+
(\varepsilon_1-\varkappa\lambda_1^{ij})u_1^{ij}
+
k f_1^{ij},
\]
\[
u_2^{i+1\,j}
=
-\varkappa \lambda_2^{ij}u_2^{i\,j+1}
+
(\varepsilon_2+\varkappa\lambda_2^{ij})u_2^{ij}
+
k f_2^{ij};
\tag{6}
\]
\[
u_1^{i+1\,\nu}
=
\alpha_1^{i+1}
+
k\sum_{p=0}^{i}\beta_1^{i+1\,p},
\qquad
u_2^{i+1\,0}
=
\alpha_2^{i+1}
+
k\sum_{p=0}^{i}\beta_2^{i+1\,p},
\]
where \(\varkappa=kh^{-1}\); \(\varepsilon_1, \varepsilon_2\) are unit matrices.
For a matrix \(A=\|a_{pq}\|\) the norm \(|A|\) is equal to \(\max_q \sum_{p=1}^{p_0}|a_{pq}|\). The matrix
\(A(t,x,u)\) satisfies the Lipschitz condition \((A\in \mathrm{Lip})\), if for every \(a>0\) there is a \(K\) such that
\[ |A(t,x,u)-A(\bar t,\bar x,\bar u)|\leq K\bigl(|t-\bar t|+|x-\bar x|+|u-\bar u|\bigr) \]
for \(|u|,|\bar u|\leq a\). If in the rectangle \(\Pi^*=\Pi_{t^*}(0\leq t\leq t^*,\,0\leq x\leq x^0)\) the integral relation
\[ \iint_{\Pi^*}\left[\left(\frac{\partial v}{\partial t}-\frac{\partial v\lambda}{\partial x}\right)u+vf\right]d\Pi = \int_{0}^{x^0}vu\Big|_{0}^{t^*}\,dx - \int_{0}^{t^*}v\lambda u\Big|_{0}^{x^0}\,dt, \]
holds for \(u,\lambda,f\in\mathrm{Lip}\), where \(v\) is any continuously differentiable row matrix, then we shall say that \(u\) is a generalized solution of system (1).
Theorem 1. If \(\lambda,f,\varphi,\alpha,\beta\) have first partial derivatives satisfying the Lipschitz condition, and the compatibility conditions (4)—(5) are fulfilled, then there exists a \(t^*>0\) such that in \(\Pi^*\) there exists a classical solution of the boundary-value problem (1)—(3), and moreover \(du/dt,\,du/dx\in\mathrm{Lip}\).
Theorem 2. If \(\lambda,f,\varphi,\alpha,\beta\in\mathrm{Lip}\) and the compatibility conditions (4) are fulfilled, then there exists a \(t^*>0\) such that in \(\Pi^*\) there exists a generalized solution of the boundary-value problem (1)—(3), and moreover \(u\in\mathrm{Lip}\).
The proofs of these theorems are based on the following lemmas.
Lemma 1. If \(\lambda,f,\varphi,\alpha,\beta\in\mathrm{Lip}\) and the compatibility conditions (4) are fulfilled, then there exist \(U_1,U_2\) and \(t^2>0\) such that, if for the mesh the relations \(\varkappa|\lambda^{ij}|\leq 1,\;0\leq a\leq \varkappa|\lambda_1^i|^{-1},\;a\leq \varkappa|\lambda_2^i|\) are fulfilled in \(\Pi_{t^1}\), then \(|u^{ij}|\leq U_1,\;|\Delta_1u^{ij}|\leq U_2,\;|\Delta_2u^{ij}|\leq U_2\) in \(\Pi_{t^*}\), where \(t^*=\min\{t^1,t^2\}\), \(\Delta_1u^{ij}=(u^{i+1,j}-u^{ij})k^{-1}\), \(\Delta_2u^{ij}=(u^{i,j+1}-u^{ij})h^{-1}\).
Lemma 2. If \(\lambda,f,\varphi,\alpha,\beta\) have first partial derivatives satisfying the Lipschitz condition, and the compatibility conditions (4)—(5) are fulfilled, then there exist \(U_3\) and \(t^3>0\) such that, if for the mesh the relations \(\varkappa|\lambda^{ij}|\leq 1,\;0<a\leq \varkappa|\lambda_1^i|^{-1},\;a\leq \varkappa|\lambda_2^i|\) are fulfilled in \(\Pi_{t^2}\), then \(|\Delta_1^2u^{ij}|\leq U_3,\;|\Delta_1\Delta_2u^{ij}|\leq U_3,\;|\Delta_2^2u^{ij}|\leq U_3\) in \(\Pi_{t^*}\), where \(t^*=\max\{t^2,t^3\}\).
Lemma 3. If \(\lambda,f\in\mathrm{Lip}\) and a sequence \(u^p\) of solutions of system (6) for meshes whose step tends to zero converges uniformly in \(\Pi^*\) to \(u\), and \(|\Delta_1u^p|\leq U_2,\;|\Delta_2u^p|\leq U_2\), then \(u\) is a generalized solution of system (1).
Theorem 3. If \(\lambda,f,\varphi,\alpha,\beta\in\mathrm{Lip}\) and the compatibility conditions (4) are fulfilled, then the generalized solution of the boundary-value problem (1)—(3) is unique.
Theorem 4. If \(\lambda,\lambda_\varepsilon,f,f_\varepsilon,\varphi,\varphi_\varepsilon,\alpha,\alpha_\varepsilon,\beta,\beta_\varepsilon\in\mathrm{Lip}\), the compatibility conditions (4) are fulfilled, the generalized solution exists in \(\Pi^*\), \(|\lambda-\lambda_\varepsilon|=O(\varepsilon),\ldots,|\beta-\beta_\varepsilon|=O(\varepsilon)\), then, for sufficiently small \(\varepsilon\), from the existence of the generalized solution \(u_\varepsilon\) in \(\Pi^*\) it follows that \(|u-u_\varepsilon|=O(\varepsilon)\).
Theorem 5. Let \(u\) be a classical solution in \(\Pi^*\) of the boundary-value problem (1)—(3), \(du/dt,\,du/dx\in\mathrm{Lip}\). If \(\lambda,f,\varphi,\alpha,\beta\in\mathrm{Lip}\), then for sufficiently small \(h\) there is a \(C\) such that from \(\varkappa|\lambda^{ij}|\leq 1\) it follows that
\[ |u(t_i,x_j)-u^{ij}|\leq Ch. \]
Theorem 6. Let \(u\) be a generalized solution in \(\Pi^*\) of the boundary-value problem (1)—(3). If \(\lambda,f,\varphi,\alpha,\beta\in\mathrm{Lip}\), then for sufficiently small \(h\) there is a \(C\) such that from \(\varkappa|\lambda^{ij}|\leq 1\) it follows that
\[ |u(t_i,x_j)-u^{ij}|\leq Ch^{1/2}. \]
The author considers it his pleasant duty to express gratitude to A. D. Myshkis for his constant attention to the present work.
Received
1 VI 1960
CITED LITERATURE
- V. E. Abolinya, On a mixed problem for linear hyperbolic systems of partial differential equations in the plane, Dissertation, Riga, 1954.
- A. Ya. Lepin, Solution of the mixed problem for hyperbolic systems in the plane by the method of grids, Dissertation, Minsk, 1955.
- V. E. Abolinya, A. D. Myshkis, Mat. sbornik, 50 (92), No. 4, 423 (1960).