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weno5.m
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%
% weno5.m
% weno5-2D
%
% Created by Yuan Leiqi (袁磊祺) on 2021/7/8.
%
function [flux, dt] = weno5(U)
% WENO5 scheme
global Nx Ny CFL dx
flux = zeros(Nx, Ny, 4); % flux vector
W = U2W(U);
f_1 = zeros(Nx, Ny, 4); % 正通量
f_2 = zeros(Nx, Ny, 4); % 负通量
g_1 = zeros(Nx, Ny, 4); % 正通量
g_2 = zeros(Nx, Ny, 4); % 负通量
h_1 = zeros(Nx, Ny, 4); % 插值后的f_1
h_2 = zeros(Nx, Ny, 4);
f_1x = zeros(Nx, Ny, 4); % f1x为正通量导数 f2x为负通量导数
f_2x = zeros(Nx, Ny, 4);
g_1y = zeros(Nx, Ny, 4); % g1x为正通量导数 f2x为负通量导数
g_2y = zeros(Nx, Ny, 4);
D_1 = zeros(4, 4); % 正特征值
D_2 = zeros(4, 4); % 负特征值
big = zeros(1, 2); % 记录通量导数最大值
Ry = ones(4, 4);
dt = CFL * dx / max(max(max(abs(W(:, :, 2)), abs(W(:, :, 3))) + sqrt(1.4 * W(:, :, 4) ./ W(:, :, 1))));
ep = 1e-6;
gam = zeros(1, 3);
gam(1) = 1/16;
gam(2) = 5/8;
gam(3) = 5/16;
beta = zeros(4, 4);
omg = zeros(3, 4);
omgm = zeros(4, 1);
q = zeros(3, 4);
for j = 1:Ny - 1
for i = 1:Nx - 1
p = W(i, j, 4); % 压强
a = sqrt(1.4 * p / U(i, j, 1)); % 声速
u = W(i, j, 2);
v = W(i, j, 3);
h = (U(i, j, 4) + p) / W(i, j, 1); % 单位质量流体热焓 《计算》P21
% x 方向
% 右特征向量矩阵
Rx = ones(4, 4);
Rx(2, 1) = u;
Rx(3, 1) = 0;
Rx(4, 1) = 0.5 * (u^2 - v^2);
Rx(1, 2) = 0;
Rx(2, 2) = 0;
Rx(4, 2) = v;
Rx(2, 3) = u - a;
Rx(3, 3) = v;
Rx(4, 3) = h - u * a;
Rx(2, 4) = u + a;
Rx(3, 4) = v;
Rx(4, 4) = h + u * a;
% 左特征向量矩阵
Lx = ones(4, 4);
b2 = 0.4 / a^2;
b1 = b2 * (u^2 + v^2) / 2;
Lx(1, 1) = 1 - b1;
Lx(2, 1) = -b1 * v;
Lx(3, 1) = 0.5 * (b1 + u / a);
Lx(4, 1) = 0.5 * (b1 - u / a);
Lx(1, 2) = b2 * u;
Lx(2, 2) = b2 * u * v;
Lx(3, 2) = -0.5 * (b2 * u + 1 / a);
Lx(4, 2) = -0.5 * (b2 * u - 1 / a);
Lx(1, 3) = b2 * v;
Lx(2, 3) = 1 + b2 * v^2;
Lx(3, 3) = -0.5 * b2 * v;
Lx(4, 3) = -0.5 * b2 * v;
Lx(1, 4) = -b2;
Lx(2, 4) = -b2 * v;
Lx(3, 4) = 0.5 * b2;
Lx(4, 4) = 0.5 * b2;
% Lx * Rx
% 对角矩阵
Gamma_x = diag([u, u, u - a, u + a]);
tempU = reshape(U(i, j, :), 4, 1);
f_1(i, j, :) = Rx * ((Gamma_x + abs(Gamma_x)) / 2) * Lx * tempU;
f_2(i, j, :) = Rx * ((Gamma_x - abs(Gamma_x)) / 2) * Lx * tempU;
% y 方向
% 右特征向量矩阵
Ry = ones(4, 4);
Ry(1, 1) = 0;
Ry(3, 1) = 0;
Ry(4, 1) = u;
Ry(2, 2) = 0;
Ry(3, 2) = v;
Ry(4, 2) = 0.5 * (v^2 - u^2);
Ry(2, 3) = u;
Ry(3, 3) = v - a;
Ry(4, 3) = h - v * a;
Ry(2, 4) = u;
Ry(3, 4) = v + a;
Ry(4, 4) = h + v * a;
% 左特征向量矩阵
Ly = ones(4, 4);
Ly(1, 1) = -b1 * u;
Ly(2, 1) = 1 - b1;
Ly(3, 1) = 0.5 * (b1 + v / a);
Ly(4, 1) = 0.5 * (b1 - v / a);
Ly(1, 2) = 1 + b2 * u^2;
Ly(2, 2) = b2 * u;
Ly(3, 2) = -0.5 * b2 * u;
Ly(4, 2) = -0.5 * b2 * u;
Ly(1, 3) = b2 * v * u;
Ly(2, 3) = b2 * v;
Ly(3, 3) = -0.5 * (b2 * v + 1 / a);
Ly(4, 3) = -0.5 * (b2 * v - 1 / a);
Ly(1, 4) = -b2 * u;
Ly(2, 4) = -b2;
Ly(3, 4) = 0.5 * b2;
Ly(4, 4) = 0.5 * b2;
% Ly * Ry
% 对角矩阵
Gamma_y = diag([v, v, v - a, v + a]);
g_1(i, j, :) = Ry * ((Gamma_y + abs(Gamma_y)) / 2) * Ly * tempU;
g_2(i, j, :) = Ry * ((Gamma_y - abs(Gamma_y)) / 2) * Ly * tempU;
end
end
% x 方向
for i = 3:Nx - 2
for j = 3:Ny - 2
for k = 1:4
beta(1, k) = (f_1(i - 2, j, k) * f_1(i - 2, j, k) * 4 - f_1(i - 2, j, k) * f_1(i - 1, j, k) * 19 + f_1(i - 1, j, k) * f_1(i - 1, j, k) * 25 + f_1(i - 2, j, k) * f_1(i, j, k) * 11 - f_1(i - 1, j, k) * f_1(i, j, k) * 31 + f_1(i, j, k) * f_1(i, j, k) * 10) / 3;
beta(2, k) = (f_1(i - 1, j, k) * f_1(i - 1, j, k) * 4 - f_1(i - 1, j, k) * f_1(i, j, k) * 13 + f_1(i, j, k) * f_1(i, j, k) * 13 + f_1(i - 1, j, k) * f_1(i + 1, j, k) * 5 - f_1(i + 1, j, k) * f_1(i, j, k) * 13 + f_1(i + 1, j, k) * f_1(i + 1, j, k) * 4) / 3;
beta(3, k) = (f_1(i + 2, j, k) * f_1(i + 2, j, k) * 4 - f_1(i + 2, j, k) * f_1(i + 1, j, k) * 19 + f_1(i + 1, j, k) * f_1(i + 1, j, k) * 25 + f_1(i + 2, j, k) * f_1(i, j, k) * 11 - f_1(i + 1, j, k) * f_1(i, j, k) * 31 + f_1(i, j, k) * f_1(i, j, k) * 10) / 3;
end
for k = 1:4
for l = 1:3
omg(l, k) = gam(l) / ((ep + beta(l, k))^2);
end
omgm(k) = omg(1, k) + omg(2, k) + omg(3, k);
end
% 三种长度为3的模版的差值结果
for k = 1:4
q(1, k) = 0.375 * f_1(i - 2, j, k) - 1.25 * f_1(i - 1, j, k) + 1.875 * f_1(i, j, k);
q(2, k) = -0.125 * f_1(i - 1, j, k) + 0.75 * f_1(i, j, k) + 0.375 * f_1(i + 1, j, k);
q(3, k) = 0.375 * f_1(i, j, k) + 0.75 * f_1(i + 1, j, k) - 0.125 * f_1(i + 2, j, k);
end
% 模版组合
for k = 1:4
h_1(i, j, k) = (q(1, k) * omg(1, k) + q(2, k) * omg(2, k) + q(3, k) * omg(3, k)) / omgm(k);
end
end
end
for i = 3:Nx - 2
for j = 3:Ny - 2
for k = 1:4
beta(1, k) = (f_2(i + 2, j, k)^2 * 4 - f_2(i + 2, j, k) * f_2(i + 1, j, k) * 19 + f_2(i + 1, j, k) * f_2(i + 1, j, k) * 25 + f_2(i + 2, j, k) * f_2(i, j, k) * 11 - f_2(i + 1, j, k) * f_2(i, j, k) * 31 + f_2(i, j, k) * f_2(i, j, k) * 10) / 3;
beta(2, k) = (f_2(i + 1, j, k) * f_2(i + 1, j, k) * 4 - f_2(i + 1, j, k) * f_2(i, j, k) * 13 + f_2(i, j, k) * f_2(i, j, k) * 13 + f_2(i + 1, j, k) * f_2(i - 1, j, k) * 5 - f_2(i - 1, j, k) * f_2(i, j, k) * 13 + f_2(i - 1, j, k) * f_2(i - 1, j, k) * 4) / 3;
beta(3, k) = (f_2(i - 2, j, k) * f_2(i - 2, j, k) * 4 - f_2(i - 2, j, k) * f_2(i - 1, j, k) * 19 + f_2(i - 1, j, k) * f_2(i - 1, j, k) * 25 + f_2(i - 2, j, k) * f_2(i, j, k) * 11 - f_2(i - 1, j, k) * f_2(i, j, k) * 31 + f_2(i, j, k) * f_2(i, j, k) * 10) / 3;
end
for k = 1:4
for l = 1:3
omg(l, k) = gam(l) / ((ep + beta(l, k))^2);
end
omgm(k) = omg(1, k) + omg(2, k) + omg(3, k);
end
% 三种长度为3的模版的差值结果
for k = 1:4
q(1, k) = 0.375 * f_2(i + 2, j, k) - 1.25 * f_2(i + 1, j, k) + 1.875 * f_2(i, j, k);
q(2, k) = -0.125 * f_2(i + 1, j, k) + 0.75 * f_2(i, j, k) + 0.375 * f_2(i - 1, j, k);
q(3, k) = 0.375 * f_2(i, j, k) + 0.75 * f_2(i - 1, j, k) - 0.125 * f_2(i - 2, j, k);
end
% 模版组合
for k = 1:4
h_2(i - 1, j, k) = (q(1, k) * omg(1, k) + q(2, k) * omg(2, k) + q(3, k) * omg(3, k)) / omgm(k);
end
end
end
for i = 4:Nx - 2
for j = 4:Ny - 2
f_1x(i, j, :) = (h_1(i, j, :) - h_1(i - 1, j, :)) / dx;
end
end
for i = 3:Nx - 3
for j = 3:Ny - 3
f_2x(i, j, :) = (h_2(i, j, :) - h_2(i - 1, j, :)) / dx;
end
end
% y 方向
for i = 3:Nx - 2
for j = 3:Ny - 2
for k = 1:4
beta(1, k) = (g_1(i, j - 2, k) * g_1(i, j - 2, k) * 4 - g_1(i, j - 2, k) * g_1(i, j - 1, k) * 19 + g_1(i, j - 1, k) * g_1(i, j - 1, k) * 25 + g_1(i, j - 2, k) * g_1(i, j, k) * 11 - g_1(i, j - 1, k) * g_1(i, j, k) * 31 + g_1(i, j, k) * g_1(i, j, k) * 10) / 3;
beta(2, k) = (g_1(i, j - 1, k) * g_1(i, j - 1, k) * 4 - g_1(i, j - 1, k) * g_1(i, j, k) * 13 + g_1(i, j, k) * g_1(i, j, k) * 13 + g_1(i, j - 1, k) * g_1(i, j + 1, k) * 5 - g_1(i, j + 1, k) * g_1(i, j, k) * 13 + g_1(i, j + 1, k) * g_1(i, j + 1, k) * 4) / 3;
beta(3, k) = (g_1(i, j + 2, k) * g_1(i, j + 2, k) * 4 - g_1(i, j + 2, k) * g_1(i, j + 1, k) * 19 + g_1(i, j + 1, k) * g_1(i, j + 1, k) * 25 + g_1(i, j + 2, k) * g_1(i, j, k) * 11 - g_1(i, j + 1, k) * g_1(i, j, k) * 31 + g_1(i, j, k) * g_1(i, j, k) * 10) / 3;
end
for k = 1:4
for l = 1:3
omg(l, k) = gam(l) / ((ep + beta(l, k))^2);
end
omgm(k) = omg(1, k) + omg(2, k) + omg(3, k);
end
% 三种长度为3的模版的差值结果
for k = 1:4
q(1, k) = 0.375 * g_1(i, j - 2, k) - 1.25 * g_1(i, j - 1, k) + 1.875 * g_1(i, j, k);
q(2, k) = -0.125 * g_1(i, j - 1, k) + 0.75 * g_1(i, j, k) + 0.375 * g_1(i, j + 1, k);
q(3, k) = 0.375 * g_1(i, j, k) + 0.75 * g_1(i, j + 1, k) - 0.125 * g_1(i, j + 2, k);
end
% 模版组合
for k = 1:4
h_1(i, j, k) = (q(1, k) * omg(1, k) + q(2, k) * omg(2, k) + q(3, k) * omg(3, k)) / omgm(k);
end
end
end
for i = 3:Nx - 2
for j = 3:Ny - 2
for k = 1:4
beta(1, k) = (g_2(i, j + 2, k)^2 * 4 - g_2(i, j + 2, k) * g_2(i, j + 1, k) * 19 + g_2(i, j + 1, k) * g_2(i, j + 1, k) * 25 + g_2(i, j + 2, k) * g_2(i, j, k) * 11 - g_2(i, j + 1, k) * g_2(i, j, k) * 31 + g_2(i, j, k) * g_2(i, j, k) * 10) / 3;
beta(2, k) = (g_2(i, j + 1, k) * g_2(i, j + 1, k) * 4 - g_2(i, j + 1, k) * g_2(i, j, k) * 13 + g_2(i, j, k) * g_2(i, j, k) * 13 + g_2(i, j + 1, k) * g_2(i, j - 1, k) * 5 - g_2(i, j - 1, k) * g_2(i, j, k) * 13 + g_2(i, j - 1, k) * g_2(i, j - 1, k) * 4) / 3;
beta(3, k) = (g_2(i, j - 2, k) * g_2(i, j - 2, k) * 4 - g_2(i, j - 2, k) * g_2(i, j - 1, k) * 19 + g_2(i, j - 1, k) * g_2(i, j - 1, k) * 25 + g_2(i, j - 2, k) * g_2(i, j, k) * 11 - g_2(i, j - 1, k) * g_2(i, j, k) * 31 + g_2(i, j, k) * g_2(i, j, k) * 10) / 3;
end
for k = 1:4
for l = 1:3
omg(l, k) = gam(l) / ((ep + beta(l, k))^2);
end
omgm(k) = omg(1, k) + omg(2, k) + omg(3, k);
end
% 三种长度为3的模版的差值结果
for k = 1:4
q(1, k) = 0.375 * g_2(i, j + 2, k) - 1.25 * g_2(i, j + 1, k) + 1.875 * g_2(i, j, k);
q(2, k) = -0.125 * g_2(i, j + 1, k) + 0.75 * g_2(i, j, k) + 0.375 * g_2(i, j - 1, k);
q(3, k) = 0.375 * g_2(i, j, k) + 0.75 * g_2(i, j - 1, k) - 0.125 * g_2(i, j - 2, k);
end
% 模版组合
for k = 1:4
h_2(i, j - 1, k) = (q(1, k) * omg(1, k) + q(2, k) * omg(2, k) + q(3, k) * omg(3, k)) / omgm(k);
end
end
end
for i = 4:Nx - 2
for j = 4:Ny - 2
g_1y(i, j, :) = (h_1(i, j, :) - h_1(i, j - 1, :)) / dx;
end
end
for i = 3:Nx - 3
for j = 3:Ny - 3
g_2y(i, j, :) = (h_2(i, j, :) - h_2(i, j - 1, :)) / dx;
end
end
for i = 4:Nx - 3
for j = 4:Ny - 3
flux(i, j, :) = f_1x(i, j, :) + f_2x(i, j, :) + g_1y(i, j, :) + g_2y(i, j, :);
end
end
end