The Error in Spatial Truncation for Systems of Parabolic Conservation Laws

• Published in: Transactions of the American Mathematical Society Vol. 311; no. 2; pp. 433 - 465
• Main Author: Kuo, Hung-Ju
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-02-1989
• Subjects: Boundary value problems; Cauchy problem; Chromatography; Conservation laws; Coordinate systems; Exact sciences and technology; Gas dynamics; Mathematics; Matrices; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, miscellaneous problems; Perceptron convergence procedure; Sciences and techniques of general use; Truncation; Parabolic equation; Conservation law; Partial differential equation; Truncation error
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-1989-0978364-X

In this paper we investigate the behavior of the solution of \begin{equation*} \begin{split} u_t = Du_{xx} - f(u)_x, \\ u(0, x) = u_0(x) \in L^\infty, & u(t, \pm L) = u^\pm, \end{split} \end{equation*} where $t \geq 0$ and $x \in \lbrack -L, L \rbrack$. Solutions of this equation are considered to be approximations to the solutions of the corresponding parabolic conservation laws. We obtain decay results on the norms of the difference between the solution for $L$ infinite and the solution when $L$ is finite.