麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 11 Hyperbolic Equations Scalar

Hyperbolic Equations Scalar One-Dimensional Conservation Laws Lecture 11

Scalar Definitions Conservation Laws Conservative Form General form (1D) au, af(a) 0 at a(e, t): is the unknown conserved quantity (mass, momentum, heat f(a): is the flux SMA-HPC⊙2003MT Hyperbolic Equations 1

Scalar Definitions Conservation Laws Primitive Form Can also be written au, af(u) au, df au 0 at aa ot du ax du du 0 at +a( a where a(u) df du Ni SMA-HPC⊙2003MT Hyperbolic Equations 2

Scalar Definitions Conservation Laws Integral Form Consider a fixed domain s≡[ar,cB]∈R (u+0f() dv=0 at dc d dt n dv=-Ifur-f(uL) SMA-HPC⊙2003MT Hyperbolic Equations 3

Scalar Derivation Example Conservation Laws Conservation of Mass Consider a volume n enclosed by surface an containing fluid of density p(a, t) and known velocity v(a, t) RATE OF CHANGE OF= MASS FLUX OF FLUID MASS INSIDE S THROUGH an 6 pdv pu nds 8g2 V·(p)dV SMA-HPC⊙2003MT Hyperbolic Equations 4

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