The convection (transport) equation

Directly to the input form

• The convection equation in one space dimension reads
  (d/dt)u + a (d/dx)u = 0,
  where we consider the region x > 0, t > 0 and a constant convection parameter a. With any smooth real function h obviously
  u(x,t) = h(x-a*t)
  is a solution. The characteristics of this equation are
  x-a*t = const
  that means lines in the (x,t) plane of the form
  t = (1/a)*x +const
  The solution u is constant along these lines. We assume that the convection parameter a satisfies a > 0, hence transport goes ''from left to right''. In order to have a well posed problem we choose as initial values
  u(x,0) = h(x) and u(0,t) = h(-a*t).
  
• As region we use [0,xend] × [0,tend] .
• We solve this problem here using finite difference methods. A necessary condition for the convergence of these methods is the so called Courant-Friedrichs-Levy condition, shortly CFL, which requires that for any computed grid point the interval(s) along the boundary, from which the value of the computed u depends (the ''numerical dependency region'') contains the intersection of the characteristic through that grid point with the initial line (either on the x or the t axis) (the ''analytical dependency region'')
• The timestep which we will use is computed as
  δ_t = tend/m
  and the space-grid as
  δ_x = xend/n
  
• We take 2n grid points for the initial values on the x-axis.
• Convection equations (which can have a much more general form, for example with a dependent on x,t,u), describe transport phenomena. In the case we consider here this is quite obvious: the initial value is shifted along the characteristics without any alteration. a > 0 has the meaning of transport speed here.
• The three methods which we present here (Friedrichs-, Lax-Wendroff- and upwind) are difference methods. Friedrichs and upwind are consistent of order one and Lax-Wendroff of order two (and convergent with this order, if CFL is satisfied)
• The formulas are:
  u_i,j+1 = (1/2)*(u_i+1,j+u_i-1,j) +a*δ_t/(2*δ_x)*(u_i+1,j-u_i-1,j) Friedrichs
  u_i,j+1 = u_i,j+ a*δ_t/δ_x*(u_i,j-u_i-1,j) Upwind
  u_i,j+1 = u_i,j+a*δ_t/(2*δ_x)*(u_i+1,j-u_i-1,j) +(1/2)*(a*δ_t/δ_x)²(u_i+1,j -2*u_i,j+u_i-1,j) Lax Wendroff
  Here i denotes the x- and j the t-grid.
• These methods approximate the solution on a grid (i*δ_x,j*δ_t) .
• CFL can be expressed here as
  a*(δ_t/δ_x) <= 1.
• Because of the quite special solution behaviour you have the chance to produce a discretization error zero here.

Input

• You can choose between the three methods.
• You specify the function h. There is a predefined case showing a wave going from left to right and you have the possibility to define h yourself. This must follow FORTRAN conventions!
• You specify a. Important: a > 0 !
• You fix the grid parameters xend, n, tend, m.
  Remember CFL !
• You can restrict the graphical output to a number of lines in x resp. t direction (gridx resp. gridt) and define your view point by the rotation angles of the x and the u axis.

Output

• You get a graphical output of the discretization error u^del - u, where ''del'' represents δ_t or equivalently δ_x.

Questions ?

• What takes place if you have a nonsmooth initial value, for example h(x) = sign(1.0e0,x) ?
• What would take place if you would violate CFL ?
• Can you verify the claimed convergence order ?

To the input form