二维波动方程的简单数值解(Matlab)

                     

贡献者: addis

  • 本词条处于草稿阶段.
预备知识 二维波动方程
图
图 1:波的干涉,动画见这里

\begin{equation} \boldsymbol{\nabla}^2 u - \frac{1}{v_0^2} \frac{\partial^{2}{u}}{\partial{t}^{2}} = 0 \end{equation}
\begin{equation} \frac{\partial^{2}{u}}{\partial{t}^{2}} = v_0^2 \boldsymbol{\nabla}^2 u \end{equation}

   使用差分法 求解常微分方程,时间差分得

\begin{equation} u(t_{i+1}) = v_0^2\Delta t^2 \boldsymbol{\nabla}^2 u(t_i) + 2u(t_{i}) - u(t_{i-1}) \end{equation}
其中,空间差分得
\begin{equation} \boldsymbol{\nabla}^2 u(x_i,y_i) = \frac{u(x_{i+1}, y_i) + u(x_{i-1}, y_i) - u(x_{i}, y_i)}{\Delta x^2} + \frac{u(x_i, y_{i+1}) + u(x_i, y_{i-1}) - u(x_i, y_{i})}{\Delta y^2} \end{equation}

   边界条件:反射,行波边界条件(类比式 5 ):

   对平面波 $u = A \sin\left( \boldsymbol{\mathbf{k}} \boldsymbol\cdot \boldsymbol{\mathbf{r}} - \omega t\right) $,有

\begin{equation} v_0 \boldsymbol\nabla u = - \hat{\boldsymbol{\mathbf{k}}} \frac{\partial z}{\partial t} \end{equation}
所以
\begin{equation} \frac{\partial z}{\partial t} = -s v_0 \left\lvert \boldsymbol\nabla u \right\rvert \end{equation}
其中 $s$ 是边界处向外法方向的方向导数的正负.例如在 $+x$ 边界处,当 $ \partial z/\partial x > 0$ 时 $s = 1$,$ \partial z/\partial x < 0$ 时 $s = -1$;在 $-x$ 边界处,当 $- \partial z/\partial x > 0$ 时 $s = 1$,$- \partial z/\partial x < 0$ 时 $s = -1$.

   以下 Matlab 代码中使用了

代码 1:wav2D.m
% 二维波动方程的简单数值解
% 使用简单的差分法
function wav2D
% === 参数设置 ====
bc = 'o'; % 边界条件: [d] Dirichlet, [n] Neumann, [o] Open
v0 = 1;
Nplot = 40;
tmin = 0; tmax = 20; Nt = 4001;
xmin = -10; xmax = 10; Nx = 800;
ymin = -10; ymax = 10; Ny = 800;
crange = [-0.4,0.4];
% ================
close all;
t = linspace(tmin, tmax, Nt); dt = (tmax-tmin)/(Nt-1);
x = linspace(xmin, xmax, Nx); dx = (xmax-xmin)/(Nx-1);
y = linspace(ymin, ymax, Ny); dy = (ymax-ymin)/(Ny-1);
% [X, Y] = ndgrid(x, y);
Z = zeros(Nx, Ny, 3);
figure; set(gcf, 'Unit', 'Normalized', 'Position', [0.1,0.1,0.4,0.4]);
for it = 1:Nt
    Z(400,400,2) = 2*sin(5*t(it));
    d2Z = laplacian(Z(:,:,2), dx, dy);
    if bc == 'd'
        Z(:,:,3) = v0^2*dt^2*d2Z + 2*Z(:,:,2) - Z(:,:,1);
    elseif bc == 'o'
        Z(2:end-1,2:end-1,3) = (v0*dt)^2*d2Z(2:end-1, 2:end-1) +...
            2*Z(2:end-1,2:end-1,2) - Z(2:end-1,2:end-1,1);
        % ymax
        grad_x = gradient(Z(:,end,2))/dx;
        grad_y = (Z(:,end,2)-Z(:,end-1,2))/dy;
        Z(:,end,3) = Z(:,end,2) ...
            - (v0*dt)*sign(grad_y).*sqrt(grad_x.^2 + grad_y.^2);
        % ymin
        grad_x = gradient(Z(:,1,2))/dx;
        grad_y = (Z(:,2,2)-Z(:,1,2))/dy;
        Z(:,1,3) = Z(:,1,2) ...
            + (v0*dt)*sign(grad_y).*sqrt(grad_x.^2 + grad_y.^2);
        % xmax
        grad_x = (Z(end,:,2)-Z(end-1,:,2))/dx;
        grad_y = gradient(Z(end,:,2))/dy;
        Z(end,:,3) = Z(end,:,2) ...
            - (v0*dt)*sign(grad_x).*sqrt(grad_x.^2 + grad_y.^2);
        % xmin
        grad_x = (Z(2,:,2)-Z(1,:,2))/dx;
        grad_y = gradient(Z(1,:,2))/dy;
        Z(1,:,3) = Z(1,:,2) ...
            + (v0*dt)*sign(grad_x).*sqrt(grad_x.^2 + grad_y.^2);
    end
    Z(:,:,1) = Z(:,:,2); Z(:,:,2) = Z(:,:,3);
    % 画图
    if mod(it, Nplot) == 0
        clf; surfCart(x, y, Z(:,:,3)); caxis(crange);
        colormap jet; shading interp; axis equal;
        xlabel x; ylabel y; view(78, 32);
        axis([xmin,xmax,ymin,ymax,-2,2]);
        title(['t = ' num2str(t(it), '%.2f')]);
        saveas(gcf, [num2str(it/Nplot) '.png']);
    end
end
end

% laplacian of Z(ix, iy)
% finite difference
% boundary condition (not included in Z) is 0
function d2Z = laplacian(Z, dx, dy)
N1 = size(Z, 1); N2 = size(Z, 2);
z1 = zeros(N1, 1); z2 = zeros(1, N2);
d2Z_1 = ([Z(2:end, :); z2] + [z2; Z(1:end-1, :)] - 2*Z)/dx^2;
d2Z_2 = ([Z(:, 2:end) z1] + [z1 Z(:, 1:end-1)] - 2*Z)/dy^2;
d2Z = d2Z_1 + d2Z_2;
end

  

未完成:这个代码应该放到别处并引用
代码 2:surfCart.m
% my version of surf()
% X, Y, Z are 2D matrices
% the color of each facet is determined by the value
%                   of the upper-left grid point
% there will be 1 facet for each element of Z,
%                    and one more grid-point at
% the end of each dimension
function h = surfCart(x,y,Z,varargin)
[X,Y] = ndgrid([x,2*x(end)-x(end-1)],[y,2*y(end)-y(end-1)]);
Z = [Z, Z(:,end); Z(end,:), Z(end,end)];
h = surf(X,Y,Z,varargin{:});
shading flat;
view(90,90);
axis([min(X(:)), max(X(:)), min(Y(:)), max(Y(:)), ...
   [-1, 1]*max(1, max(abs(Z(:))))]);
xlabel x; ylabel y; zlabel z;
colorbar;
set(datacursormode(gcf), 'UpdateFcn', @datatip);
cameratoolbar;
end

                     

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