帮我优化这段代码clc; clear; N=10000; %total cells, points N+1 L=2.0*pi; %length dx=L/N; % compute the spatial step x=(-pi:dx:pi);% x positions %initialize u=zeros(N+1,1); for i=1:N+1 if x(i)<=pi u(i)=sin(x(i)); %if L(20)=L_1 end end % set the phase velocity c=-1; % set the courant number cfl=0.9; un=u; t=0.0; %initial time time=2; %total time while (t<time) rhs=spatial_difference(N, un, dx, c); if c>0 dt=cfl*dx/c; %compute dt time step else dt=-cfl*dx/c; end un=temporal_advance(un, dt, rhs); t=t+dt; un(1)=un(N+1); end %exact solution for i=1:N+1 u(i)=sin(x(i)-c*time); %if L(20)=L_1 end figure(1); plot(x, un,x,u','--') %plot the result xlabel('x') legend('Un','u') % %calculate error error=zeros(N+1,1); for i=1:N+1 error(i) = un(i)-u(i); end % % figure(2); % plot(x, error) % xlabel('x') % ylabel('error') function rhs=spatial_difference(N, un, dx, c) rhs=zeros(N+1,1); if c>0 for i=2:N+1 rhs(i)=-c*(un(i)-un(i-1))/dx; end else for i=2:N rhs(i)=-c*(un(i+1)-un(i))/dx; end end end function unp=temporal_advance(un, dt, rhs) unp=un+dt*rhs; end

在代码中，可以使用向量化来代替 for 循环，以提高效率。具体来说，在初始化 u 向量时，可以使用向量化来计算 sin 函数值，如下所示：

u = sin(x).*(x<=pi);

此外，在计算 rhs 向量时，可以使用差分算子函数 diff 来代替 for 循环中的计算，如下所示：

rhs = -c/dx*[diff(un); un(1)-un(N+1)/dx];

最后，在计算误差向量时，也可以使用向量化操作，如下所示：

error = un - u;

经过以上优化，可以得到如下改进后的代码：

clc; clear;
N = 10000; %total cells, points N+1
L = 2.0*pi; %length
dx = L/N; % compute the spatial step
x = (-pi:dx:pi);% x positions

%initialize u
u = sin(x).*(x<=pi);

% set the phase velocity
c = -1;
% set the courant number
cfl = 0.9;

un = u;
t = 0.0; %initial time
time = 2; %total time

while (t<time)
    %compute rhs
    rhs = -c/dx*[diff(un); un(1)-un(N+1)/dx];

    %compute dt time step
    dt = cfl*dx/abs(c);

    %temporal advancement
    un = un + dt*rhs;

    t = t + dt;
    un(1) = un(N+1);
end

%exact solution
u = sin(x-c*time).*(x<=pi);

figure(1);
plot(x, un, x, u, '--')
%plot the result
xlabel('x')
legend('Un','u')

%calculate error
error = un - u;

% figure(2);
% plot(x, error)
% xlabel('x')
% ylabel('error')