Numerical methods for conservation laws, HW 1

\[\left\{ \begin{array}{ll} u_{t} + au_{x} = 0 \\ u(x, 0) = u_{0}(x)\\ u(0, t) = u(1, t) \end{array} \right.\]

on \((x, t) \in [0, 1] \times [0, 5]\)

Your code should take spatial stepsize ∆x, a, \(\lambda = a \Delta t\), and \(u_0(x)\) as input.

The output is a video showing the solution as time t goes from 0 to 5.

\[\vec{u}_{t} + f(\vec{u})_{x} + g(\vec{u})_{y} = 0.\]

This conservation law is always hyperbolic if \(u\) is scalar. If u is a vector, ,then the equation is called hyperbolic conservation law if, for any real \(\varepsilon = (\varepsilon _{1}, \varepsilon_{2}) \) the linear combination of the Jacoians \(\varepsilon _{1} f'(u) + \varepsilon _{2}g'(u)\) has only eigenvalues and complete set of eigenvectors.

Euler equations of compressible gas dynamics.

\[u + f(u)_{x} + g(u)_{y} = 0

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