无粘burgers方程的精确解

无粘burgers方程的精确解

function Burgers_Exact_Solution
F = @(x) sin ( x ); dF = @(x ) cos ( x );
psi = @(x ,t , xi ) x - F( xi )* t - xi ;
dpsi = @(x ,t , xi ) - dF ( xi )* t - 1 ;
N = 100;
x = linspace ( 0, 2*pi , N );
Output ( x , 0, F( x ) )
t = 0 ; Step = 0;
T = 1 ; % 中断时间为 1
N_T = 20; dt = T/ N_T ;
while ( t <= T )
Step = Step + 1; t = t + dt ;
U = Exact_Solution ( x , t ) ;
Output ( x , t , U )
end
% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
function U = Exact_Solution ( x , t )U = zeros ( size ( x ) );for i = 1: Nxi = Newton ( x(i ), t );U(i ) = F( xi );end
end
function xi = Newton ( x , t )
K_Max = 100000;
k = 1 ;
xi0 = x ;
tol = 1e-12;
r = 1 ;
while (r >tol && k <= K_Max )
xi = xi0 - ...
psi ( x , t , xi0 ) /...
dpsi ( x , t , xi0 ) ;
r = abs ( xi - xi0 ) ;
xi0 = xi ;
k = k + 1 ;
end
end
% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
function Output ( x , t , U )
plot ( x , U , 'r -o' )
axis ( [0 , 2*pi , -1, 1 ] )
title(['t=',num2str(t)]);
pause ( 1 )
end
end