Increased Productivity Example #9

Burgers' PDE Equation & Parameter Estimation

Burgers' PDE ExampleBurgers' Equation, a non-linear PDE, occurs as a model for a number of physical problems (e.g. Fluids, Heat, Traffic, Shock Waves, etc.). The equation is Ut + Ux U = v Uxx, where U = U(x,t), Ux = Partial of U w.r.t. X, & v = viscosity.

Using a calculus level language, we'll show how to solve such a PDE in a minimum of time while tweaking parameter(s) to meet user's requirements. Most math models with PDEs and/or ODEs may be solved in a week. Solution accuracy improves due to the behind the scene use of Automatic Differentiation (AD). For a mental picture of what's going on, think of AD as taking symbolic derivatives of the equations involved in your model and evaluating these derivatives at given points. Derivative accuracy is thus as accurate as your computer allows.

The key calculus level statements are Find & Integrate; Find locates 'best' user parameters in order to meet an objective. For example,

Find a, Ut0     ooo       to match error
is requesting to vary parameters a & Ut0 ooo until error = 0.

Integrate does exactly that for given variables stated in ones Initiate statement. For example,

initiate Gemini for PDE equations Ut/U of T ooo


integrate PDE; by Gemini

Optimization is next once satisfied with solving a PDE or system of PDEs. Code wise, converting to optimization just requires another Find statement wrapped 'around' the present code.

An example optimization problem with Burgers' Equation is found in Optimal Control for fluid flow. The problem is to determine the most inexpensive control that will produce a flow to match a given target. Solution: Add 1) the user parameters that can be varied in your model, 2) objective function & 3) outer Find statement. Then you are ready to solve your optimization problem. Now tweak, tweak, tweak until experience corrects your math model and objective function for your problem. (I'm speaking from experience; one job/problem took some two years to solve! Our math model and objective function had to be modified and modified and ... modified.)

Increased Productivity Example #9 Source Code:

      global all
      problem BurgersPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: Burgers' 1D Equation; a PDE; Parameter 
C --- Estimation; Initial Value Problem solved using Method of Lines.
C ------------------------------------------------------------------------
        dynamic U, Ut, Ut0
C User parameters ...
        viscosit=.01        ! viscosity between .1 & .001 are of interest
        ipoints=20          ! grid pts. over x-axis
        tFinal= .5          ! not sure when odd numeric problem surfaces
        jpoints=10*tFinal   ! grid pts. over t-axis
C x-parameter initial settings: x ==> i
        ip=ipoints:   a=.5:    pi= 4*atan(1.)
        xFinal= 1:    dx=xFinal/ipoints
        allot U(ip), Ut(ip), Ut0(ip)

C t-parameter initial settings: t ==> j
        tPrint= tFinal/jpoints:    dt=tPrint / 10
        print 78, "viscosity=", viscosit, dt
 78     format( 1x, a, f5.2, 2x, f5.3)
        Find a   in xAxis   by Ajax   to match error
      model xAxis
C ... Integrate over x-axis

C settings at x=0
        do 1 ii=1, ipoints
          xPrt=ii * dx:      call U0( xPrt)
          Ut(ii)= Ut0(ii):     U(ii)= U0start
 1      continue

        t= 0:   tPrt=tPrint:  dt=tPrt / 10
        print 79, t, (U(ii),ii=1,ip)
!        Initiate athena;  for PDE;
        Initiate ISIS;  for PDE;
     ~       equations Ut/U;  of t;  step dt;  to tPrt
        do while (t .lt. tFinal)
          Integrate PDE;  by ISIS
          if( t .ge. tPrt) then
            print 79, t, (U(ii),ii=1,ip)
            tPrt=tPrt + tPrint
            if( t-dt .le. 0.2 .and. 0.2 .lt. t+dt) error= (a - U(15))**2
          end if
        end do
 79     format( 1x, f5.2, 1x, 20(1pg12.4))
      model PDE                         ! Partial Differential Equation
C                                       ! Method of Lines
        Vxx=(U(3) - 2*U(2) - U(1))/dx**2
        Ut(1)= viscosit * Vxx - (U(2)-U(1))/dx
        do 20 ii=2,ipoints-1         ! System of ODEs
          Vxx=(U(ii+1) - 2*U(ii) - U(ii-1))/dx**2
          Ut(ii)= viscosit * Vxx - (U(ii+1)-U(ii-1))/(2*dx)
 20     continue
        Vxx=(U(ipoints) - 2*U(ipoints-1) - U(ipoints-2))/dx**2
        Ut(ipoints)= viscosit * Vxx - (U(ipoints)-U(ipoints-1))/dx
      Fmodel U0(xx)       ! Initial starting values @ t = 0
        if( xx .le. 0) then
          U0 = 0
        elseif( xx .lt. .5 ) then
          U0 = (1 - cos( 4 * pi * xx))/2
          U0 = 0

Solving PDEs are another increased productivity example do to using Calculus (level) Programming.

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