Scalar Derivation Example Conservation Laws Conservation of Mass dv=0 n l at holds for all so we can write 8P+V·(p)=0 t This is the differential form of the conservation law SMA-HPC⊙2003MT Hyperbolic Equations 5
Scalar Examples Conservation Laws Linear Advection Equation Model convection of a concentration p(a, t) op, apa dp 0 at a at aa a: constant SMA-HPC⊙2003MT Hyperbolic Equations 6
Scalar Examples Conservation Laws Inviscid Burgers" Equation Flux function f(u) Conservation law 0u0u2 0 ot a at da N3 SMA-HPC⊙2003MT Hyperbolic Equations 7
Scalar Examples Conservation Laws Traffic Flow Let p(a, t) denote the density of cars (vehicles/km) and a(a, t) the velocity. Since cars are conserved dp, apu 0 ot ax Assume that u is a function of p a(p)= umax Pmax where0≤p≤ Pmax and umax is some maximum speed( the speed limit N4 SMA-HPC⊙2003MT Hyperbolic Equations 8
Scalar Examples Conservation Laws Buckley-Leverett Equation Consider a two phase(oil and water )fluid flow in porous medium.Let0≤u(x,t)≤1 represent the saturation of water au af(u) 0 at dr f(u)= u2+a(1-u) a: constant 1 SMA-HPC⊙2003MT Hyperbolic Equations 9