I have decided to teach myself some CFD coding using my own intuition and some free online tutorials. This is a post about code I wrote to solve a first order ordinary differential equation. The method employed is called the finite difference method. I will start by coding equations using this method and will slowly move to finite volume method. I hope this code helps you in learning something.
The equation in question is mentioned below.
dy/dx = x+4 and y(0)=3
A uniform grid was employed, as I have not yet figured out how to make non uniform grid. When I figure it out, I will update it here, which will be soon.
The code, mentioned below, starts with a variable called Nx, which represents the total number of grid points in which the domain is divided in to. A rule of thumb, the more points in the domain, more accurate the solution becomes at the cost of computation resources, mainly CPU time and RAM and storage.
x and y are the vectors in which the known and the unknown values are stored. x has the known values, like the length of a pipe through which a fluid flow or a rod through which heat flows etc. y stores the unknown values like temperature of the rod or velocity of the fluid etc. The values are stored at nodes.
y(1) means first value in the vector y. which also happens to be our boundary condition. y is the value of the function at x=0. x=0 is the first value in in the x vector, written as x(1).
x(Nx-1) represents the domain size. Minimum domain size is usually zero e.g. starting point of a rod or a pipe and maximum can be any real number.
To calculate the grid spacing, represented by dx, the domain size was divided by the number of elements. In this example, we have four nodes, two at corners and two in-between, forming a total of three elements.
First order finite difference method was used to discretize the given equation. And a for loop was used for the solution. Using 4 grid points, the solution has an accuracy of 2.22% against the analytical solution of the equation which can be verified by hand calculations. Thank you for reading.
%solution to dy/dx=x+4 y(0)=3
%uniform grid
clear; clc;
Nx=4; %number of grid points
x=zeros(Nx,1); %x vector, known, goes from some known value to maximum value of the domain
y=zeros(Nx,1); %y vector, one value known due to boundary condition rest unknown
y(1)=3; %boundary condition
x(Nx-1)=1; %domain size
dx=x(Nx-1)/(Nx-1); %grid spacing
for i=1:Nx-1
x(i+1)=(i)*dx;
y(i+1)=(dx.*(x(i)+4))+y(i); %discretized differential equation
end
The equation in question is mentioned below.
dy/dx = x+4 and y(0)=3
A uniform grid was employed, as I have not yet figured out how to make non uniform grid. When I figure it out, I will update it here, which will be soon.
The code, mentioned below, starts with a variable called Nx, which represents the total number of grid points in which the domain is divided in to. A rule of thumb, the more points in the domain, more accurate the solution becomes at the cost of computation resources, mainly CPU time and RAM and storage.
x and y are the vectors in which the known and the unknown values are stored. x has the known values, like the length of a pipe through which a fluid flow or a rod through which heat flows etc. y stores the unknown values like temperature of the rod or velocity of the fluid etc. The values are stored at nodes.
y(1) means first value in the vector y. which also happens to be our boundary condition. y is the value of the function at x=0. x=0 is the first value in in the x vector, written as x(1).
x(Nx-1) represents the domain size. Minimum domain size is usually zero e.g. starting point of a rod or a pipe and maximum can be any real number.
To calculate the grid spacing, represented by dx, the domain size was divided by the number of elements. In this example, we have four nodes, two at corners and two in-between, forming a total of three elements.
First order finite difference method was used to discretize the given equation. And a for loop was used for the solution. Using 4 grid points, the solution has an accuracy of 2.22% against the analytical solution of the equation which can be verified by hand calculations. Thank you for reading.
%solution to dy/dx=x+4 y(0)=3
%uniform grid
clear; clc;
Nx=4; %number of grid points
x=zeros(Nx,1); %x vector, known, goes from some known value to maximum value of the domain
y=zeros(Nx,1); %y vector, one value known due to boundary condition rest unknown
y(1)=3; %boundary condition
x(Nx-1)=1; %domain size
dx=x(Nx-1)/(Nx-1); %grid spacing
for i=1:Nx-1
x(i+1)=(i)*dx;
y(i+1)=(dx.*(x(i)+4))+y(i); %discretized differential equation
end
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