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

Using a calculus level language, such as Fortran Calculus (FC), We'll show how to solve such a PDE in a minimum of time while improving a solutions accuracy. 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.

At present, PDEs must be reduced to a system of ODEs in order to solve them with the Fortran Calculus compiler. The Method of Lines (MOL) is used to solve Burgers' 1D PDE as an example (code below or see PDE-ivp.fc source file). We hope to remove this restriction once enough interest in solving PDEs with Fortran Calculus is shown.

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

find Ux0, Uxx0 ooo to minimize gx

is requesting FC to vary parameters Ux0 & Uxx0 until gx is minimized.

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

initiate Isis; for xAxis; Equations Uxx/Ux, Ux/U; of x; step dx; to xP ooo
integrate xAxis; by Isis

statements are seeking to integrate the PDE variables (U, Ux, & Uxx) over x. Then the outer find statement will vary initial starting values (Ux0 & Uxx0) until gx is minimized.

Optimization is next once satisfied with solving a PDE or system of PDEs. The hardest part is often deciding on what are the primary objective goals. Code wise, converting to optimization just requires another Find statement put '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.)

      global all
      problem BurgersPDE
C ------------------------------------------------------------------------
C --- FORTRAN CALCULUS Application: Burgers' 1D Equation; a PDE Initial
C --- Value Problem.
C
C ... Warning ... a numerical problem exists with this code when 'dt' or
C ...   'viscosit' values are too small.
C ------------------------------------------------------------------------
        dynamic U, Ux, Uxx

C User parameters ...
        viscosit = .1         ! viscosity between .1 & .001 are of interest
        iPoints = 100         ! grid pts. over x-axis
        jPoints = 10          ! grid pts. over t-axis
        tFinal =  1           ! not sure when odd numeric problem surfaces

C x-parameter initial settings: x ==> i
        xFinal =  1:    ip = ipoints
        xPrint = xFinal/ipoints  :  dx = xPrint / 10

C t-parameter initial settings: t ==> j
        pi= 4*atan(1):     jp = jpoints:    dt = tFinal/jpoints
        allot U( jp), Ux( jp), Uxx( jp)
                      
C settings/guesses at t = 0
        do 10 jj = 1, jpoints
          U(jj) = 0:      Ux(jj) = 0:     Uxx(jj) = 0
 10     continue

        call plotit
        call xAxis
	close( 31)
      end

C ... Integrate over x-axis
      model xAxis
 15     format( 1x, 1Pg12.6, 2x, 10(G16.8, 2x))
 79     format( 1h , a, 2x, f8.4, 3x, f8.4, 3x, (9g14.5, 3x))
        initiate ISIS;  for PDE;
     ~       equations Uxx/Ux, Ux/U;  of x;  step dx;  to xp
        x= 0
        Do 10 ii = 1, ipoints
          xp = ii * xPrint
          integrate PDE;  by ISIS
          write(31,15) x, (U( jj), jj = 1, jp)
          print 79, "x,U's= ", x, (U( jj), jj = 1, jp)
 10     continue
      end
      model PDE        		! Partial Differential Equation
        Uxx(1) = U0ivp(x)		! Method of Lines
        do 20 jj = 2, jpoints		! System of ODEs
          Uxx(jj)= ((U(jj) - U(jj-1))/dt + U(jj) * Ux(jj))/viscosit
 20     continue
      end
      Fmodel U0ivp(xx)       ! Initial starting value for t-axis integration
        if( xx .lt. 0) then
          U0ivp = 0
        elseif( xx .le. .5) then
          U0ivp = (1 - cos( 4 * pi * xx))/2
        else
          U0ivp = 0
        endif
      end
      procedure plotit
        character*8 date, time, string*80
 11     format( 3X, 2f11.6, 5x, a)
        call getdate( date, time)
	open( 1, file='CON')          ! Output to CONSOLE
        write(1,*) 'Enter a viscosity value: '
        read *, viscosit
        open( 31, FILE='~4plot3d.dat', Status='unknown')
        write(6,11) dt, viscosit, ' = dt & viscosity'
        string = "Burgers' 1D PDE Solution on " // date // "  @  "
        string = charnb( string) // time
        write(31, *) string
        write(31,11) dt, viscosit, ' = dt & viscosity'
        write(31, *) iPoints, jPoints, '      iPoints & jPoints'
        write(31, *) " PDE:  Ut = -U Ux + viscosit Uxx"
        write(31, *) " -------------------------------"
        write(31, *) " "
        write(31, *) " Solved PDE using 'Method of Lines'"
      end
      subroutine getdate( date1, time1)
        character*8  date1, time1
        call date( date1)
        call time( time1)
        return
      end

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