Wave Equation Using Matlab and Finite Differencing.

Question

Tyler Reohr 2021-11-18

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1589589-wave-equation-using-matlab-and-finite-differencing

回答： Neelanshu 2024-4-15

在 MATLAB Online 中打开

I'm trying to solve a wave equation,

, over

and

,with boundary/initial conditions:

I'm relatively new to numerically solving PDEs, but I'm not entirely sure what I'm doing wrong here. My code yields that the string vibrates to a height on the order of

units, which is obviously unreasonable given the initial values. The spatial (n) / time (m) intervals are 25 and 700 respectively for the final equation, however they are shortened to 15 each in order to help me diagnose the code.

Here's what I have so far:

clear all;close all;clc;

% Initialize Variables

n = 15; % spacial steps

m = 15; % time steps

Li = 0;

Lf = 2;

ti = 0;

tf = 50;

h = Lf/n;

k = tf/m;

u_left = 0;

u_right = 0;

u0 = @(x) -(x-1)^2 + 1;

ut0 = @(x) (x-1)^4 -1;

v = ut0;

c = sqrt(.25);

u = zeros(m+1,n+1);

% Boundary Conditions

u(:,1) = u_left;

u(:,n+1) = u_right;

% Initial Conditions

for i=2:n

u(1,i) = u0(i*h); % Dirichlet Condition

end

for i=2:n

u(2,i) = u0(i*h) + ut0(i*h)*k; % Niemann Condition, used to fill timestep 2

end

% Solve using Finite Differencing

for i=3:m

u(i,2:n) = c*k/h^2*u(i-1,1:n-1) + (1-2*c*k/h^2) * u(i-1,2:n) + c*k/h^2*u(i-1,3:n+1);

end

format long

u

u = 16×16

1.0e+31 * 0 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0 0 0 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 0 0 0 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 -0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 -0.000000000000000 0 0 -0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 0 0 0.000000000000000 -0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000000 0 0 -0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 0 0 0.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000000 0 0 -0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 0 0 0.000000000000004 -0.000000000000005 0.000000000000004 -0.000000000000002 0.000000000000001 -0.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000001 0.000000000000002 -0.000000000000004 0.000000000000006 -0.000000000000004 0 0 -0.000000000001124 0.000000000001568 -0.000000000001274 0.000000000000701 -0.000000000000266 0.000000000000067 -0.000000000000011 0.000000000000016 -0.000000000000094 0.000000000000353 -0.000000000000895 0.000000000001580 -0.000000000001903 0.000000000001347 0

surf(u)

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Answer 1

Neelanshu 2024-4-15

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1589589-wave-equation-using-matlab-and-finite-differencing#answer_1441326

在 MATLAB Online 中打开

Hi Tyler,

I found some errors in the equations for the initial condition and the finite difference method that you used to solve the wave equation above.

Instead of (ck/h^2), it should be ((ck/h)^2) in the finite difference method. Furthermore, it seems that there are certain terms missing in the finite difference equation. In the initial conditions, when using u0 and ut0, the input argument should correspond to (i-1) instead of (i). Kindly refer to the attached code below for your reference:

clear all;close all;clc;

%%

% Initialize Variables

n = 25; % spacial steps

Li = 0; %inital

Lf = 2; %final

h = Lf/n; %total number steps x

m = 700; % time steps

ti = 0; %initial

tf = 50; %final

k = tf/m; %total number steps time

u_left = 0;

u_right = 0;

u0 = @(x) -(x-1)^2 + 1; %u(x,0)

ut0 = @(x) (x-1)^4 -1; %d/dt(u(t)) @ (x,0)

v = ut0;

c = sqrt(.25);

u = zeros(m+1,n+1); %u(t,x)

%%

% Boundary Conditions

u(:,1) = u_left; %u(0,t) x(0)==u(:,1)

u(:,n+1) = u_right; %u(2,t) 2==n+1

%%

% Initial Conditions

for i=2:n

%i*h misses the first coordinate

u(1,i) = u0((i-1)*h); % Dirichlet Condition

end

for i=2:n

%i*h misses the first coordinate

u(2,i) = u0((i-1)*h) + ut0((i-1)*h)*k; % Niemann Condition, used to fill timestep 2

end

% Solve using Finite Differencing

r = (c*k/h)^2;

for i=3:m

%u(i,2:n) = c*k/h^2*u(i-1,1:n-1) + (1-2*c*k/h^2) * u(i-1,2:n) + c*k/h^2*u(i-1,3:n+1);

u(i,2:n) = 2*u(i-1,2:n) - u(i-2,2:n) + r*(u(i-1,3:n+1) - 2*u(i-1,2:n) + u(i-1,1:n-1));

end

format long

u

u = 701x26

0 0.153600000000000 0.294400000000000 0.422400000000000 0.537600000000000 0.640000000000000 0.729600000000000 0.806400000000000 0.870400000000000 0.921600000000000 0.960000000000000 0.985600000000000 0.998400000000000 0.998400000000000 0.985600000000000 0.960000000000000 0.921600000000000 0.870400000000000 0.806400000000000 0.729600000000000 0.640000000000000 0.537600000000000 0.422400000000000 0.294400000000000 0.153600000000000 0 0 0.133342354285714 0.258533668571429 0.374801554285714 0.481443840000000 0.577828571428571 0.663394011428571 0.737648640000000 0.800171154285714 0.850610468571429 0.888685714285714 0.914186240000000 0.926971611428571 0.926971611428571 0.914186240000000 0.888685714285714 0.850610468571429 0.800171154285714 0.737648640000000 0.663394011428571 0.577828571428571 0.481443840000000 0.374801554285714 0.258533668571428 0.133342354285714 0 0 0.111460218775510 0.220888911486881 0.325284741224490 0.423243365131195 0.513500874635569 0.594933795451895 0.666559087580175 0.727534145306123 0.777156797201166 0.814865306122449 0.840238369212828 0.852995117900875 0.852995117900875 0.840238369212828 0.814865306122449 0.777156797201166 0.727534145306122 0.666559087580175 0.594933795451895 0.513500874635568 0.423243365131195 0.325284741224490 0.220888911486880 0.111460218775510 0 0 0.089173203230202 0.182241112515024 0.274485002894032 0.363508069949426 0.447414450824062 0.524518934089368 0.593346959745344 0.652634619220563 0.701328655372166 0.738586462485869 0.763776086275957 0.776476223885286 0.776476223885286 0.763776086275956 0.738586462485869 0.701328655372166 0.652634619220563 0.593346959745344 0.524518934089368 0.447414450824061 0.363508069949426 0.274485002894032 0.182241112515024 0.089173203230202 0 0 0.067662396640294 0.143429087836491 0.223043359405010 0.302753027043494 0.379972419230083 0.452454587393348 0.518233451533290 0.575623800221336 0.623221290600345 0.659902448384602 0.684824667859820 0.697426211883144 0.697426211883144 0.684824667859820 0.659902448384602 0.623221290600345 0.575623800221336 0.518233451533290 0.452454587393348 0.379972419230083 0.302753027043494 0.223043359405010 0.143429087836491 0.067662396640294 0 0 0.047766763550846 0.105383880037009 0.171620728206717 0.241501676051645 0.311586266139109 0.379054282465639 0.441448125031237 0.496661279550187 0.542938317451062 0.578874895876718 0.603417757684298 0.615864731445231 0.615864731445231 0.603417757684298 0.578874895876718 0.542938317451062 0.496661279550187 0.441448125031237 0.379054282465639 0.311586266139109 0.241501676051645 0.171620728206717 0.105383880037009 0.047766763550846 0 0 0.029834290724335 0.069056571568591 0.120924360495992 0.180290910647037 0.242678633902541 0.304642702473969 0.363231698392337 0.415917804514786 0.460594506555602 0.495576593086214 0.519600155535194 0.531822588188255 0.531822588188255 0.519600155535194 0.495576593086214 0.460594506555602 0.415917804514785 0.363231698392337 0.304642702473969 0.242678633902541 0.180290910647037 0.120924360495992 0.069056571568591 0.029834290724335 0 0 0.013772829959352 0.035249493505774 0.071722484419444 0.119682260417902 0.173686567935887 0.229558475668472 0.283838834852145 0.333578067505206 0.376318573627655 0.410094273219492 0.433430606280716 0.445344532811329 0.445344532811329 0.433430606280717 0.410094273219492 0.376318573627655 0.333578067505206 0.283838834852145 0.229558475668472 0.173686567935887 0.119682260417902 0.071722484419444 0.035249493505773 0.013772829959352 0 0 -0.000753268563310 0.004431160533687 0.024809907028480 0.060278276068776 0.105066711833432 0.154157055673220 0.203540931745106 0.249843495010412 0.290255968479800 0.322531403581841 0.344984588887963 0.356492050112453 0.356492050112453 0.344984588887963 0.322531403581841 0.290255968479800 0.249843495010413 0.203540931745106 0.154157055673220 0.105066711833432 0.060278276068776 0.024809907028479 0.004431160533687 -0.000753268563310 0 0 -0.014095993030586 -0.023358968237627 -0.019095331685527 0.002731766752406 0.037304219425797 0.078814136202541 0.122628927712011 0.164935036634736 0.202571664039983 0.233010974593919 0.254357105439400 0.265346148004998 0.265346148004998 0.254357105439400 0.233010974593919 0.202571664039983 0.164935036634736 0.122628927712011 0.078814136202541 0.037304219425797 0.002731766752406 -0.019095331685527 -0.023358968237628 -0.014095993030586 0

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surf(u)

I hope this helps.