11 can be determined by diagonalizing the jacobian

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Jacobian (10.1.2b). This can be done for hyperbolic systems since A(u) has m distinct eigenvalues (De nition 10.1.1). Thus, let P = p(1) p(2) : : : p(m) ] (10.1.6a) and recall the eigenvalue-eigenvector relation AP = P (10.1.6b) 4 Hyperbolic Problems where 2 6 =6 6 4 3 7 7 7 5 1 2 ... (10.1.6c) m Multiplying (10.1.2a) by P;1 and using (10.1.6b) gives P;1ut + P;1Aux = P;1ut + P;1ux = P;1b: Let w = P;1u so that (10.1.7) wt + wx = P;1ut + (P;1)tu + P;1ux + (P;1)xu]: Using (10.1.7) wt + wx = Qw + g (10.1.8a) where Q = (P;1)t + (P;1)x]P g = P;1b: (10.1.8b) i = 1 2 : : : m: (10.1.8c) In component form, (10.1.8a) is (wi)t + i(wi)x = m X j =1 qi j wj + gi Thus, the transformation (10.1.7) has uncoupled the di erentiated terms of the original system (10.1.2a). Consider the directional derivative of each component wi, i = 1 2 : : : m, of w, dwi = (w ) + (w ) dx i = 1 2 ::: m it ix dt dt in the directions dx = i = 1 2 ::: m (10.1.9a) dt i and use (10.1.8c) to obtain m dwi = X q w + g i = 1 2 : : : m: (10.1.9b) dt j=1 i j j i 10.1. Conservation Laws 5 The curves (10.1.9a) are called the characteristics of the system (10.1.1, 10.1.2). The partial di erential equations (10.1.2) may be solved by integrating the 2m ordinary differential equations (10.1.9a, 10.1.9b). This system is uncoupled through its di erentiated terms but coupled through Q and g. This method of solution is, quite naturally, called the method of characteristics. While we could develop numerical methods based on the method of characteristics, they are generally not e cient when m > 2. De nition 10.1.2. The set of all points that determine the solution at a point P (x0 t0) is called the domain of dependence of P . Consider the arbitrary point P (x0 t0 ) and the characteristics passing through it as shown in Figure 10.1.2. The solution u(x0 t0 ) depends on the initial data on the interval A B ] and on the values of b in the region APB , bounded by A B ] and the characteristic curves x = 1 and x = m . Thus, the region APB is the domain of dependence of P . _ _ t P(x 0 ,t 0) 11...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern