Computational Methods for Hyperbolic Conservation Laws

where u : R × R → R is a vector of conserved variables (or state variables). For fluid dynamics, u is the vector of mass, momentum and energy denisties so that ∫ b a uj(x, t) dx is the total quantity of the j state variable in the interval at time t. Because these variables are conserved, ∫∞ −∞ uj(x, t) dx should be constant in t. The function f : R m → R is the flux function, which gives the rate of flow of the conserved variables at any point. The PDEs must be augmented by initial data, u(x, 0) = u0(x). Then the equation (1) and this initial data constitute the Cauchy problem. If (1) holds only on an interval [a, b] ∈ R, boundary data must be specified. We assume (1) is hyperbolic. That is, we assume the m×m Jacobian matrix A = f ′ = df du has m real eigenvalues λj(u), j = 1, . . . ,m and a complete set of linearly independent eigenvectors. This means that A is diagonalizable. The system is called strictly hyperbolic if these eigenvalues are distinct. In two space dimensions, the conservation law takes the form