Smooth Total Derivative Solutions Recall the primitive form of the conservation law ou 0u_0 at +a(u) da The total time variation of u(a, t), on an arbitrary curve a=a(t), in the s-t plane, is d ou d ou dt ot dt aa SMA-HPC⊙2003MT Hyperbolic Equations 10
Smooth Characteristics Solutions dar dt a(u)→ dt 0→u=uo( constant) da The curves a =c(t), such that dt=a(u) are called characteristics constant→a(u) constant→ characteristics are straight lines SMA-HPC⊙2003MT Hyperbolic Equations 11
Smooth Characteristics Solutions t=0 0 dm=a(0)→2=0+a(uo)t N6 SMA-HPC⊙2003MT Hyperbolic Equations 12
Smooth Examples Solutions Linear Advection Equation t↑dm Solution p(a, t) pola-at Characteristic lines p(a =0+at SMA-HPC⊙2003MT Hyperbolic Equations 13
Smooth Examples Solutions Burgers Equation Recall f(u)=u2, so a(u) 8x0 0u_0 at Solution :a(, t) mola- ut The solution is constant along the characteristic lines defined by a - ut =a SMA-HPC⊙2003MT Hyperbolic Equations 14