欧拉方程 godunov方法 matlab代码

欧拉方程是一类描述气体动力学的偏微分方程，而Godunov方法是一种数值求解偏微分方程的方法。下面是使用MATLAB实现Godunov方法求解一维欧拉方程的示例代码：

% 定义计算区间
x_start = 0;
x_end = 1;
t_start = 0;
t_end = 0.2;

% 定义网格参数
nx = 100; % 空间网格数
nt = 1000; % 时间网格数
dx = (x_end - x_start) / nx; % 空间步长
dt = (t_end - t_start) / nt; % 时间步长

% 定义初始条件
rho_l = 1;
rho_r = 0.125;
u_l = 0;
u_r = 0;
p_l = 1;
p_r = 0.1;

% 初始化计算区域
x = linspace(x_start + 0.5 * dx, x_end - 0.5 * dx, nx);
rho = zeros(1, nx);
u = zeros(1, nx);
p = zeros(1, nx);

% 设置初始条件
for i = 1:nx
    if x(i) <= 0.5
        rho(i) = rho_l;
        u(i) = u_l;
        p(i) = p_l;
    else
        rho(i) = rho_r;
        u(i) = u_r;
        p(i) = p_r;
    end
end

% 计算气体常数
gamma = 1.4;
c = sqrt(gamma * p ./ rho);

% 计算初始通量
flux = zeros(3, nx);
flux(1, :) = rho .* u;
flux(2, :) = rho .* u.^2 + p;
flux(3, :) = u .* (gamma * p ./ (gamma - 1) + 0.5 * rho .* u.^2 + p);

% 进行时间推进
for n = 1:nt
    % 计算相邻网格之间的通量
    f = zeros(3, nx - 1);
    for i = 1:nx - 1
        if u(i) > c(i)
            f(:, i) = flux(:, i);
        elseif u(i+1) < -c(i+1)
            f(:, i) = flux(:, i+1);
        else
            rho_star = (p(i+1) - p(i) + rho(i) * u(i) * (c(i) - u(i+1)) - rho(i+1) * u(i+1) * (c(i+1) + u(i+1))) / ((c(i) - u(i+1)) + (rho(i) * (c(i) - u(i+1)) - rho(i+1) * (c(i+1) + u(i+1))) / rho(i+1));
            u_star = (u(i) * (rho(i) * (c(i) - u(i+1)) - rho(i+1) * (c(i+1) + u(i+1))) + (p(i+1) - p(i))) / ((c(i) - u(i+1)) + (rho(i) * (c(i) - u(i+1)) - rho(i+1) * (c(i+1) + u(i+1))) / rho(i+1));
            p_star = p(i) + rho(i) * (u_star - u(i)) * (c(i) - u(i));
            f(1, i) = rho(i) * u(i);
            f(2, i) = rho(i) * u(i)^2 + p(i);
            f(3, i) = u(i) * (gamma * p(i) / (gamma - 1) + 0.5 * rho(i) * u(i)^2 + p(i));
            if u_star > 0
                f(1, i) = f(1, i) + rho(i) * (u_star - u(i));
                f(2, i) = f(2, i) + rho(i) * (u_star - u(i)) * u_star + p_star - p(i);
                f(3, i) = f(3, i) + u_star * (gamma * p_star / (gamma - 1) + 0.5 * rho(i) * (u_star^2 - u(i)^2 + 2 * c(i) * (u_star - u(i))));
            else
                f(1, i) = f(1, i) + rho_star * (u_star - u(i+1));
                f(2, i) = f(2, i) + rho_star * (u_star - u(i+1)) * u_star + p_star - p(i+1);
                f(3, i) = f(3, i) + u_star * (gamma * p_star / (gamma - 1) + 0.5 * rho_star * (u_star^2 - u(i+1)^2 + 2 * c(i+1) * (u_star - u(i+1))));
            end
        end
    end
    
    % 计算时间推进后的解
    rho_new = rho;
    u_new = u;
    p_new = p;
    for i = 2:nx-1
        rho_new(i) = rho(i) - dt / dx * (f(1, i-1) - f(1, i));
        u_new(i) = u(i) - dt / dx * (f(2, i-1) - f(2, i)) / rho_new(i);
        p_new(i) = p(i) - dt / dx * (f(3, i-1) - f(3, i));
    end
    
    % 更新通量
    rho = rho_new;
    u = u_new;
    p = p_new;
    flux(1, :) = rho .* u;
    flux(2, :) = rho .* u.^2 + p;
    flux(3, :) = u .* (gamma * p ./ (gamma - 1) + 0.5 * rho .* u.^2 + p);
end

% 绘制结果
figure;
subplot(3, 1, 1);
plot(x, rho);
xlabel('x');
ylabel('rho');
subplot(3, 1, 2);
plot(x, u);
xlabel('x');
ylabel('u');
subplot(3, 1, 3);
plot(x, p);
xlabel('x');
ylabel('p');

在这个示例代码中，我们使用Godunov方法求解了一维欧拉方程，并绘制了密度、速度和压强随空间位置的变化。