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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 I'm new to Method of Lines (MOL) and PDEs, so please check my work!

C Internet shows solutions to Burgers Equation that differ greatly!!!
C Are the solutions different types; e.g. Steady State vs. zzz?

C ------------------------------------------------------------------------
C --- Calculus Programming example: Burgers' 1D Equation; a PDE Initial
C --- Value Problem solved using Method of Lines.
C ------------------------------------------------------------------------
C ... Warning ... a numerical problem exists when 'dt' or 'viscosit' values
C ...   are too small.
C ------------------------------------------------------------------------
        dynamic U, Ut, error
C User parameters ...
        viscosit = 1         ! viscosity between .1 & .001 are of interest
        tFinal =  .5          ! not sure when odd numeric problem surfaces
        jpoints = 10*tFinal  ! grid pts. over t-axis
C x-parameter initial settings: x ==> i
        xFinal= 1
	 dx = .1
        ipoints = xFinal/dx + 1.99        ! grid pts. over x-axis
        allot U( ipoints), Ut( ipoints), error( ipoints)

C t-parameter initial settings: t ==> j
        dt = .005:	tPrint= dt*jpoints:  pts = ipoints
        print 78, "viscosity, dt, dx, ipoints =", viscosit,dt,dx,pts
 78     format( 1x, a, f5.3, 20(2x, f8.4))
        call xAxis
      model xAxis
C ... Integrate over x-axis ... for a steady state solution
	last = 5	! number of iterations for Steady State solution
	do 10 i = 1, last
        t= 0:   tPrt = tPrint:  dt = tPrt / 10
        ‹error› = ‹U›
        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) .and. (i .eq. 1)) then
	     print 79, t, (U( j), j = 1, ipoints)
            tPrt = tPrt + tPrint
          elseif((t .ge. tPrt) .and. (i .eq. last)) then
	     print 79, t, (U( j), j = 1, ipoints)
            tPrt = tPrt + tPrint
          elseif(t .ge. tPrt) then
	     print 78, "-------------", i
            print 78, " "
            tPrt = tPrt + tPrint
        end do
 10	continue
        ‹error› = ‹U› - ‹error›
	 print 78, " "
	 print 78, "i & 'error' array follows = ", i
        print 79, t, (error( j), j = 1, ipoints)
	 print 78, "-------------"
	 print 78, " "
 78     format( 1x, a, i8)
 79     format( 1x, f4.3, 20(1x, f8.5))
      model PDE                         ! Partial Differential Equation
C                                       ! Method of Lines
C Boundary Conditions for Burgers' Equation
	 U(1) = 1/(1 + Exp(-t/(4*viscosit))
        U(ipoints) = 1/(1 + Exp((2-t)/(4*viscosit))

	 do 20 jj = 2, ipoints-1       ! System of ODEs ... Method of Lines
	   Ut(jj) = -U(jj) * (U(jj+1)-U(jj-1))/(2*dx)
	   Ut(jj) = Ut(jj) + viscosit*(U(jj+1)-2*U(jj)+U(jj-1))/(dx*dx)
 20     continue

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

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<a href=""><img style="float:left; width:100px" src="http://goal-driven.net/image/fc-compiler-icon.png"/> <strong>Burgers Partial Differential Equation</strong> </a>; Simulation to Optimization, Tweak Parameters for Optimal Solution.

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