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Hyperbolic Equations

Partial Differential Equation Toolbox™ solves equations of the form

m2ut2+dut·(cu)+au=f

When the d coefficient is 0, but m is not, the documentation calls this a hyperbolic equation, whether or not it is mathematically of the hyperbolic form.

Using the same ideas as for the parabolic equation, hyperbolic implements the numerical solution of

m2ut2(cu)+au=f

for x in Ω, where x represents a 2-D or 3-D point, with the initial conditions

u(x,0)=u0(x)ut(x,0)=v0(x)

for all x in Ω, and usual boundary conditions. In particular, solutions of the equation utt - cΔu = 0 are waves moving with speed c.

Using a given mesh of Ω, the method of lines yields the second order ODE system

Md2Udt2+KU=F

with the initial conditions

Ui(0)=u0(xi)iddtUi(0)=v0(xi)i

after we eliminate the unknowns fixed by Dirichlet boundary conditions. As before, the stiffness matrix K and the mass matrix M are assembled with the aid of the function assempde from the problems

–∇ · (cu) + au = f and –∇ · (0∇u) + mu = 0.

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