我正在尝试在 MATLAB 中实现 KdV 方程的有限差分方案，并且我已经准备好了大部分代码，除了使用初始条件在第一级进行近似。有人建议我使用欧拉的方法在 m=1 处获得 'u'，然后使用 m>=2 的方案。在这种情况下如何应用欧拉方法？即使只是在第一级近似的一般答案也将不胜感激。我包含我的代码以供参考
close all
clear
clc
% Generating grid with n points, with the space between two points being
%(x2-x1)/(n-1)
x = linspace(-5,5,1001);
N=1001;
h=x(2)-x(1); % grid size
dt=0.05;
%Soliton initial condition
Am=8; %Amplitude
mu=sqrt(Am/2);
x0=-15;
c=1;
syms H(x)
H(x)=piecewise(x < 0,0,x > 0,1);
u= Am*(sech(mu*(x'-x0))).^2+c^2*H(x);
% Creating a matrix A - First order
A = diag(ones(N-1,1),1)-diag(ones(N-1,1),-1);
Cvector = zeros(N, 1);
Cvector(end) = 1;
u_ic = Cvector;
% First order finite difference scheme
diff_first=A*u/(2*h)+1/(2*h)*u_ic;
% Weighted average matrix for the term 'u'
A_w = diag(ones(N-1,1),1)+diag(ones(N-1,1),-1)+diag(ones(N,1));
diff_w=2*A_w*u +2*u_ic;
% Matrix multiplication of first derivative and weighted average for 6uu_x
diff_middle=diff_first.*diff_w;
% Creating a Third Order Matrix
r = zeros(1,N);
r(2:3) = [-2,1];
c = -r;
A_third = toeplitz(c,r);
% Difference scheme for third order term
diff_third=A_third*u/(h*h*h)-1/(h*h*h)*u_ic;
%Computing finite difference method
u = u - 2*dt*diff_middle-dt*diff_third;
plot(u)