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Hyperbolic Equations

Hyperbolic Equations. IVP:. u ( x ,0) = f ( x ), -  < x < . IBVP:. u ( x ,0) = f ( x ), 0 < x <  and u (0, t ) = g ( t ), t > 0. IVP. Method of Characteristics. Let C be a characteristic curve described by x = x ( t ). On C u ( x , t ) = u ( x ( t ), t ).

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Hyperbolic Equations

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  1. Hyperbolic Equations IVP: u(x,0) = f(x), - < x <  IBVP: u(x,0) = f(x), 0 < x <  and u(0,t) = g(t), t > 0

  2. IVP

  3. Method of Characteristics • Let C be a characteristic curve described by x = x(t). • On Cu(x,t) = u(x(t), t). • Differentiating u along C results in the equation:

  4. Method of Characteristics • a is defined as the wave speed on the curve C • The change of u with respect to t on C is zero • the first equation is solved to obtain

  5. Characteristics • given the initial condition • the solution to the (kinematic) wave equation is example...

  6. Courant-Friedrichs-Lewy (CFL) Condition The numerical domain of dependence must contain the analytical domain of dependence, where the analytical domain of dependence is given by the characteristic curves. How does this govern the relationship between the time step and space step, given the wave speed a?

  7. CFL Condition

  8. Hyperbolic IBVPs • Explicit methods shown previously work, with addition of the BC at x = 0 • Implicit methods are unconditionally stable • Some implicit methods do not require solving a system of equations!

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