Solving partial differential finite difference

Hi there,

I am trying to solve the wave equation in polar coordinates using finite difference method. The image below is the equation/finite difference. I tried to follow the method in here but I am not sure how to change x and y to theta and r on MATLAB. Any help would be greatly appriciated.

Thanks!

I

Answer 1

[Irina Ayelen Jimenez]

clear;clc;

RUN = 1;

% Domain

R = 2;

% number of elements

nx = 20; ny = 50;

dx = R/nx; dy = 2*pi/ny;

r = linspace(0.01,R,nx); phiy = linspace(0,2*pi,ny);

[Rx,phi] = meshgrid(r,phiy);

xx = r.*cos(phi); yy = r.*sin(phi);

%%% Variables

wn = zeros(nx,ny); % current

wnm1 = wn; % time n-1

wnp1 = wn; % time n+1

%%% Parameters

CFL = 0.5; % CFL = c.dt/dx

c = 1;

% dt = CFL*dx/c;

dt = dx*dy/(4*c);

% dt=0.01;

% choose steps

steps = 3000;

%%% Initial condition peak

% wn(3:6,2:7) = 1;

% initial condition 1st mode

m = 0;n = 1;

km = [0 1.842 3.052 4.2012;3.8317 5.33 6.70 8.0152; 7.0156 8.5363 9.9695 11.346];

kb=km(n+1,m+1);

p = cos(m*phi).*besselj(m,kb*Rx/R);

wn = p';

wnp1(:) = wn(:);

t = 0;

figure;(1)

tn = 1;

% wn = p; wnp1 = p;

if RUN == 1

for k = 1:steps

% Reflecting boundary condition

wn(end,:) = 0;

% wn(1,1) = 0;

% Solution

t = tn*dt;

wnm1 = wn; wn = wnp1; % save current and previus array

% Source

% wn(10,:) = dt^2*20*sin(10*pi*t/20);

% wn(:,ny) = wn(:,1);

for i = 2:nx-1, for j = 2:ny-1

ri = max([r(i),0.5*dx]);

wnp1(i,j) = c*dt^2*((wn(i+1,j)-2*wn(i,j)+wn(i-1,j))/dx^2 + 1/ri*(wn(i+1,j)-wn(i-1,j))/dx/2 + 1/ri^2*(wn(i,j+1)-2*wn(i,j)+wn(i,j-1))/dy^2) ...

+ 2*wn(i,j)-wnm1(i,j);

end, end

% for j = 2:ny-1

% i = nx;

% wnp1(i,j) = c*dt^2*(1/ri^2*(wn(i,j+1)-2*wn(i,j)+wn(i,j-1))/dy^2);

% end

wnp1(1,:) = wnp1(2,:); % center

wnp1(:,1) = wnp1(:,2); wnp1(:,ny) = wnp1(:,1); % angular continuity

clf;

surf(xx,yy,wn');title(sprintf('t = %.2f', t));

zlim([-2 2])

shg;pause(0.01)

Matrix(:,:,tn) = wn;

tn = tn + 1;

end

end

[aktham mansi]

https://www.mathworks.com/matlabcentral/answers/1691530-finite-difference-method-for-laplace-equation-in-polar-form

can you help me?