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1 introduction

To help learn a little more about Ada, I wrote ¹ the following small program which solves the advection pde \(u_t + a u_x=0\) with initial condition \(u(x,0)=\sin (2\pi x)\) and with periodic boundary conditions \(u(0,t)=u(1,t)\).

To plot the solution, the output of the program is directed to a text file, then the the text file was loaded into Matlab to ran the simulation.

The program contains 3 files

1. main.adb the main line.
2. lax_wendroff_pkg.ads the package interface.
3. lax_wendroff_pkg.adb the package body.

These are the steps to compile the program

$ gnatmake -gnat05  -gnatwa main.adb 
gcc -c -gnat05 -gnatwa lax_wendroff_pkg.adb 
gnatbind -x main.ali 
gnatlink main.ali
 

To run the program do ./main > result.txt

Matlab was used to plot the solution as follows

A=load('result.txt','-ascii'); 
B=reshape(A,100,100); 
 
for i=1:size(B,2) 
    plot(B(:,i)) 
    ylim([-1 1]); 
    drawnow(); 
 
    pause(0.01); 
end
 

2 Ada source code listing

2.1 main.adb

-- compile with gnatmake -gnataw main.adb 
-- Main driver to solve u_t + speed * u_x = 0 
-- the advection PDE with periodic boundary 
-- conditions and with initial conditions sin(2*pi*x) 
-- 
-- Using Ada 2005 OO features 
-- For illustration and learning only. 
-- 
-- By Nasser M. Abbasi 
-- Match 26, 2011 
 
with Ada.Text_IO; use Ada.Text_IO; 
with Lax_Wendroff_pkg; Use Lax_Wendroff_pkg; 
with Ada.Numerics; use Ada.Numerics; 
with Ada.Numerics.Generic_Elementary_Functions; 
 
procedure main is 
package Math is new Ada.Numerics.Generic_Elementary_Functions(Float); 
use Math; 
 
h       : constant float := 0.01; -- grid space 
courant : constant float := 0.8;  -- CFL condition 
speed   : constant float := 1.0;  -- flow speed 
k       : constant float := courant*h/speed;  -- time step size 
L       : constant float := 1.0;  -- domain length 
N       : constant positive := natural(L/h)+1;  --number of grid points 
 
-- generate initial condition 
function initialize(h: float; N: natural) return solution_t is 
       data : solution_t(1..N) := (others=>0.0); 
       x: float:=0.0; 
       begin 
         for i in data'range LOOP 
             data(i) := sin(2.0*pi*x); 
             x := x + h; 
         end LOOP; 
         RETURN(data); 
       end; 
 
-- create the Lax-Wendroff object 
o    : lax_Wendroff_t := make(speed=>speed , h=>h,k =>k, u0 =>initialize(h,N)); 
 
-- to print the final solution 
procedure print_solution(o: lax_wendroff_t) is 
   u: constant solution_t := o.get_solution; 
   begin 
      FOR i in u'range LOOP 
          put_line(float'image(u(i))); 
      END LOOP; 
 
end; 
 
begin 
 
      -- run for 100 steps for now 
      FOR i in 1..100 LOOP 
          o.step; 
          print_solution(o); 
      END LOOP; 
 
end main;

2.2 lax_wendroff_pkg.ads

-- Package spec for Lax Wendroff scheme to solve 1-D 
-- Advection PDE in Ada 2005 
-- by Nasser M. Abbasi 
-- 
with Ada.Numerics.Generic_Real_Arrays; 
package Lax_Wendroff_pkg is 
   type Lax_Wendroff_t (<>) is tagged private; 
   type solution_t is array (Natural range <>) of Float; 
 
   -- primitive operations, constructor 
   function make(speed,h,k : Float; u0 : solution_t) return  Lax_Wendroff_t; 
   procedure step (o : in out Lax_Wendroff_t); 
   function get_solution (o : Lax_Wendroff_t) return solution_t; 
 
private 
   package My_Vectors is new Ada.Numerics.Generic_Real_Arrays (Float); 
   use My_Vectors; 
   subtype buffer_t is Real_Vector; 
 
   type Lax_Wendroff_t (N : Positive) is tagged record 
      speed       : Float;                -- speed of flow 
      h           : Float;                -- space grid size 
      k           : Float;                -- delt 
      u           : buffer_t (-1 .. N);   -- solution 
      step_number : Natural; 
   end record; 
 
end Lax_Wendroff_pkg;

2.3 lax_wendroff_pkg.add

-- Package body for Lax Wendroff scheme to solve 1-D 
-- Advection PDE in Ada 2005 
-- by Nasser M. Abbasi 
-- 
 
package body Lax_Wendroff_pkg is 
 
   -- constructor 
   function make 
     (speed : Float; 
      h     : Float; 
      k     : Float; 
      u0    : solution_t) 
      return  Lax_Wendroff_t 
   is 
      o : Lax_Wendroff_t (u0'Length); 
   begin 
 
      o.speed                  := speed; 
      o.h                      := h; 
      o.k                      := k; 
      o.step_number            := 0; 
      o.u (0 .. u0'Length - 1) := buffer_t (u0); 
      o.u (o.u'First)          := 0.0; 
      o.u (o.u'Last)           := 0.0; 
 
      return (o); 
   end make; 
 
   -- make solution step 
   procedure step (o : in out Lax_Wendroff_t) is 
      package my_vectors is new Ada.Numerics.Generic_Real_Arrays (Float); 
      use my_vectors; 
      u : buffer_t renames o.u; 
      a : constant Float := o.speed * o.k / o.h; 
      subtype r is Natural range o.u'First + 1 .. o.u'Last - 1; 
      subtype r_plus_1 is Natural range r'First + 1 .. r'Last + 1; 
      subtype r_minus_1 is Integer range r'First - 1 .. r'Last - 1; 
 
   begin 
      o.step_number := o.step_number + 1; 
 
      u (r) := u (r) - 
               (a / 2.0) * (u (r_plus_1) - u (r_minus_1)) + 
               (a ** 2) / 2.0 * 
               (u (r_minus_1) - 2.0 * u (r) + u (r_plus_1)); 
 
      -- adjust due to periodic boundary conditions 
      u (u'First) := u (u'Last - 2); 
      u (u'Last)  := u (u'First + 2); 
 
   end step; 
 
   -- get the current solution 
   function get_solution (o : Lax_Wendroff_t) return solution_t is 
   begin 
      return solution_t (o.u (o.u'First + 1 .. o.u'Last - 1)); 
   end get_solution; 
 
end Lax_Wendroff_pkg;