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