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2D_Heat_Equation-MPI

Introduction

2D Heat Equation for a bottle. Solution found by parallelising the linear system of algebraic equations with MPI. The bottle has an intial temperature distrubution: data for this can be found in bottle.dat. The end result after running the application will give the temperature at a certain point in the distribution.

Scenario

The diagram below shows the general form taken by a 2D heat distribution:

                        I = 0
                        
           [0][0]------------------[0][N-1]
                |                  |
         J = 0  |                  |  J = N-1
                |                  |
         [M-1][0]------------------[M-1][N-1]
          
                        I = M-1

This diagram will prove very useful so refer to it. Solution to the heat-equation:

  u(Central) = (1/4)*([u(North] + u[South] + u[East] + u[West])

Operating System

Ubuntu 20.04

Text Editor

Any Text Editor will do but I've used Visual Studio Code. Vim, gedit, Emacs etc.

Compiler

gcc/g++ 9.4.0

Background: Parallel Programming

The parallel strategy used is Domain Decomposition. The problem (global domain) is decomposed into smaller domains (processes). "Workers" in the subdomains perform the calculations and then communicate the results with the master (global domain). This link provides a basic example behind the Master/Slave concept.

Using MPI

MPIcan be downloaded here or at the command line (see below).

$ sudo apt-get install mpich

Running the Application

Once the application has been compiled, use the MPI script, mpirun script along the the number of processors you wish to use at the command line. Note: there are a few options. That is, the initial temperature distribution can be added, spacial dimementions of the temperature distribution and the and the number of time steps. Below shows that 4 processors are used, the initial temperature data is being used along with a 800 x 800 grid with 1000 time steps.

  • $ git clone http://github/MRLintern/2D_Heat_Equation-MPI.git
  • $ make
  • $ mpirun -np ./main_heat bottle.dat 800 800 1000