Documentation

This is machine translation

Translated by Microsoft
Mouse over text to see original. Click the button below to return to the English verison of the page.

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.

Was this topic helpful?