The wave equation

Directly to the input form

• The wave equation is the most prominent example of a second order hyperbolic differential equation. The variant which we consider here has one space dimension and reads
  (d/dt)**2 u = c**2 (d/dx)**2 u
  with x in ]0,1[ and t > 0. The characteristics of this equation are the lines
  t = (1/c)*x + a₁ , t= -(1/c)*x + a₂
  with arbitrary constants a₁, a₂. c has the meaning of speed of propagation.
• This equation describes the simplest form of a wave propagation.
• In order to get a well posed problem one needs boundary conditions for the boundaries x=0, t >= 0 and x=1, t >= 0 and initial conditions for the initial displacement u(x,0) and is speed (d/dt)u(x,t), t=0. The present implementation fixes the boundary values at
  u(x,t)=0, x in {0,1}
  and also the initial speed to
  (d/dt)u(x,t)_t=0=0.
  This has the advantage that the exact solution is known analytically: take the odd periodic continuation of u(x,0) over the complete real axis. Given a point (x,t) with 0 <= x <= 1, t > 0 draw the two characteristics through this point and follow them to their intersection with the x-axis. Take the midvalue of the function values of this continuation at these points: this is equal to u(x,t). Hence given an arbitrary u(x,0) (of course with u(0,0)=u(1,0)=0) we know the discretization error of any method we might try.
• you may think of u as the normal displacement of a string of length one, fixed at its ends and of (d/dt)u as its initial normal displacement speed.
• Here we use the standard 5 point difference discretization of the differential operator. This is equivalent to using the vertical method of lines on the x-grid and then Störmers discretization method for a second order ordinary differential equation in explicit form.
• The scheme reads
  u^h_j,n+1 = 2*u^h_j,n-u^h_j,n-1
  +((Δ_t/Δ_x)*c)**2*(u^h_j+1,n-2*u^h_j,n+ u^h_j-1,n)
  n denotes the time step and j the number in the space grid.
• The discretization is second order consistent in Δ_x, Δ_t and convergent if the Courant-Friedrichs-Levy condition is satisfied, which boils down here to
  c*Δ_t/Δ_x < = 1 .
  

Input

• There are three predefined cases for u(x,0) but you also have the possibility to define the initial displacement yourself.
• In this latter case you need to specify
  1. A title text for the graphical output.
  2. the initial displacement u(x,0) as a piece of code following FORTRAN conventions.
• You specify the wave speed c. Important: c >0 !
• The number n of grid points along the x-axis including the two boundary values: Important 3<=n<=2000 !
• The endtime tend of the computation. Important: tend > 0 !
• The number m of time steps. This implies
  Δ_t= tend/m.
  It is necessary to satisfy the Courant-Friedrichs-Levy condition! That is
  ((n-1)/m)tend <= 1/c .
  
• The number eachx of vertical lines in t-direction for the plot. Important : 1 <= eachx <= (n-1)/2 !
• The number eacht of horizontal lines for the plot. Important : 1 <= eacht <= (m-1)/2 !
• You specify your view point via two rotation angles for the x and the u-axis.

Output

• You get two plots: one with the computed solution and one with the discretization error on the required grid. More precisely you get the piecewise linear interpolation of u^h which may look strange if you use a crude grid.

Questions?!

• Can you verify the statement on the convergence order by varying n and m ?
• What takes place if you use a ''tricky'' combination of n,m,c ? Think about the role of the characteristics!
• What appears if you use a nonsmooth initial displacement or one which is incompatible with the boundary conditions?

To the input form