Doing Numerics in Matlab

[skrieger]

1. Homework Statement

This is for a course in numerical solutions of PDEs. So far it has all been theory of PDEs which is great since I am the worst programmer in the world. Right now I have to program the following: to solve the simplest hyperbolic equation,

du/dt = du/dx,

using the scheme

v(_j)(^n+1) = (1 + kD_0)v(_j)(^n)
v(_j)(^0) = sin(x_j)

for h(x-mesh size) = (2*pi)/(N+1), k(t-mesh size) = 2h/3, total time T = [0, 2*pi], in the three cases N = 20, 40 and 80.

the idea being to examine the errors between this result and the analytic solution, which should increase exponentially.


2. The attempt at a solution

I prefer to learn this in Matlab, but my knowledge base does not extend very far at all beyond plugging things into ode45. I just need someone to help me figure out how to set up basic difference schemes, I don't get the code beyond setting up linspaces, etc. So far all I have is

N = 32;
M = 20;
t = linspace(0,(2*pi),N)';
dt = t(2) - t(1);
x = linspace(0, (2*pi),M)';
dx = x(2) - x(1);
% Create the first derivative operator
D1 = diag(ones(N,1));
D2 = diag(-ones(N,1),-1);
D2 = D2(1:N,1:N);
D = (D1+D2)/((2*pi)/21);

but even if this is the proper formation of the derivative operator I'm not sure how to set up the actual scheme from here. No need for anyone to solve for the three N's or talk about errors, I just need help learning how to put these schemes together.

Thanks!

 

[mighty2000]

Thank you for reaching out for help with your numerical solutions of PDEs course. It's great to hear that you have a strong understanding of the theory of PDEs, but may need some assistance with programming. I understand the importance of being able to effectively code in order to solve complex problems.

Based on the information you provided, it seems like you are on the right track with setting up the basic difference scheme. Your code for creating the first derivative operator looks correct, but I would suggest double checking to make sure it is accurately representing the equation you are trying to solve.

To set up the actual scheme, you will need to use a loop to iterate through the time steps and use the derivative operator to update the values of your variable v at each spatial point. This can be done by using the formula you provided in your post: v(_j)(^n+1) = (1 + kD_0)v(_j)(^n). You will also need to specify the initial conditions for v(_j)(^0) = sin(x_j).

I understand that you prefer to use Matlab for this problem, so I would suggest looking into the "for" loop function in Matlab to help you iterate through the time steps and update your variable v. Additionally, there are many online resources and tutorials available for coding difference schemes in Matlab that may be helpful to you.

I hope this helps guide you in the right direction. Don't be discouraged if you run into some challenges along the way, as programming can be a complex and iterative process. Just remember to break down the problem into smaller steps and don't hesitate to seek out additional resources or assistance from your instructor or classmates.

Best of luck with your project!