DO j = 2, ny-1 rho_new(i,j) = rho(i,j) - dt/dx*u(i,j)*(rho(i,j)-rho(i-1,j))- dt/dy*v(i,j)*(rho(i,j)-rho(i,j-1))& + 0.5*dt/dx*u(i,j)*(1.0-u(i,j)*dt/dx)*(rho(i+1,j)-rho(i,j))- 0.5*dt/dx*u(i,j)*(1.0+u(i,j)*dt/dx)*(rho(i,j)-rho(i-1,j))& + 0.5*dt/dy*v(i,j)*(1.0-v(i,j)*dt/dy)*(rho(i,j+1)-rho(i,j))- 0.5*dt/dy*v(i,j)*(1.0+v(i,j)*dt/dy)*(rho(i,j)-rho(i,j-1))
时间: 2023-06-23 07:08:38 浏览: 35
这是一个二维的离散化的非压缩流体连续性方程,用于求解流体的密度分布。其中 rho_new 是新时刻的密度分布,rho 是当前时刻的密度分布,u 和 v 分别是 x 和 y 方向的速度分量,dt 是时间步长,dx 和 dy 分别是 x 和 y 方向的网格大小。这个方程描述了流体在时间上的演化过程,通过对速度和密度的计算,可以求解出流体在下一个时刻的状态。
相关问题
对于考虑重力源项的二维无黏欧拉方程,特别针对Rayleigh-Taylor 不稳定性问题,如何利用二阶MUSCL进行空间离散,利用二阶runge kutta方法进行时间离散,请给出fortran代码示例
下面是一个示例Fortran代码,用于求解考虑重力源项的二维无黏欧拉方程,使用了二阶MUSCL空间离散和二阶Runge-Kutta时间离散:
```
program main
implicit none
integer, parameter :: nx = 100, ny = 100
real, parameter :: dx = 0.01, dy = 0.01, dt = 0.001
real, parameter :: g = 9.8
real :: x(nx), y(ny), u(nx,ny), v(nx,ny), rho(nx,ny), p(nx,ny)
real :: u_l(nx,ny), u_r(nx,ny), v_l(nx,ny), v_r(nx,ny), rho_l(nx,ny), rho_r(nx,ny), p_l(nx,ny), p_r(nx,ny)
real :: u_t(nx,ny), v_t(nx,ny), rho_t(nx,ny), p_t(nx,ny)
real :: u_star(nx,ny), v_star(nx,ny), rho_star(nx,ny), p_star(nx,ny)
real :: u_new(nx,ny), v_new(nx,ny), rho_new(nx,ny), p_new(nx,ny)
integer :: i, j, n
! Initialize grid
do i = 1, nx
x(i) = (i-0.5)*dx
end do
do j = 1, ny
y(j) = (j-0.5)*dy
end do
! Initialize variables
do i = 1, nx
do j = 1, ny
if (y(j) >= 0.5) then
rho(i,j) = 1.0
p(i,j) = 1.0
else
rho(i,j) = 2.0
p(i,j) = 2.0
end if
u(i,j) = 0.0
v(i,j) = 0.0
end do
end do
! Time loop
do n = 1, 10000
! Save old values
u_new = u
v_new = v
rho_new = rho
p_new = p
! First RK step
do i = 2, nx-1
do j = 2, ny-1
! Calculate left and right states using MUSCL
call muscl(u_l(i,j), u_r(i,j), u(i-1,j), u(i,j), u(i+1,j))
call muscl(v_l(i,j), v_r(i,j), v(i,j-1), v(i,j), v(i,j+1))
call muscl(rho_l(i,j), rho_r(i,j), rho(i-1,j), rho(i,j), rho(i+1,j))
call muscl(p_l(i,j), p_r(i,j), p(i-1,j), p(i,j), p(i+1,j))
! Calculate fluxes
call flux(u_t(i,j), v_t(i,j), rho_t(i,j), p_t(i,j), u_l(i,j), u_r(i,j), v_l(i,j), v_r(i,j), rho_l(i,j), &
rho_r(i,j), p_l(i,j), p_r(i,j), g)
! Calculate intermediate values
u_star(i,j) = u(i,j) - dt/dx*u_t(i,j)
v_star(i,j) = v(i,j) - dt/dy*v_t(i,j)
rho_star(i,j) = rho(i,j) - dt/dx*rho_t(i,j)
p_star(i,j) = p(i,j) - dt/dx*p_t(i,j)
end do
end do
! Second RK step
do i = 2, nx-1
do j = 2, ny-1
! Calculate left and right states using MUSCL
call muscl(u_l(i,j), u_r(i,j), u_star(i-1,j), u_star(i,j), u_star(i+1,j))
call muscl(v_l(i,j), v_r(i,j), v_star(i,j-1), v_star(i,j), v_star(i,j+1))
call muscl(rho_l(i,j), rho_r(i,j), rho_star(i-1,j), rho_star(i,j), rho_star(i+1,j))
call muscl(p_l(i,j), p_r(i,j), p_star(i-1,j), p_star(i,j), p_star(i+1,j))
! Calculate fluxes
call flux(u_t(i,j), v_t(i,j), rho_t(i,j), p_t(i,j), u_l(i,j), u_r(i,j), v_l(i,j), v_r(i,j), rho_l(i,j), &
rho_r(i,j), p_l(i,j), p_r(i,j), g)
! Calculate new values
u_new(i,j) = 0.5*(u(i,j) - dt/dx*u_t(i,j) + u_star(i,j) - dt/dx*u_t(i,j))
v_new(i,j) = 0.5*(v(i,j) - dt/dy*v_t(i,j) + v_star(i,j) - dt/dy*v_t(i,j))
rho_new(i,j) = 0.5*(rho(i,j) - dt/dx*rho_t(i,j) + rho_star(i,j) - dt/dx*rho_t(i,j))
p_new(i,j) = 0.5*(p(i,j) - dt/dx*p_t(i,j) + p_star(i,j) - dt/dx*p_t(i,j))
end do
end do
! Update values for next time step
u = u_new
v = v_new
rho = rho_new
p = p_new
end do
contains
subroutine muscl(a_l, a_r, a_im1, a_i, a_ip1)
real, intent(in) :: a_im1, a_i, a_ip1
real, intent(out) :: a_l, a_r
real :: a_im, a_ip, da_im, da_ip, da_i
real :: alpha, beta
a_im = 2.0*a_i - a_im1
a_ip = 2.0*a_i - a_ip1
da_im = a_i - a_im1
da_ip = a_ip1 - a_i
da_i = 0.5*(da_im + da_ip)
alpha = minmod(2.0*da_i, da_im, da_ip)
beta = minmod(2.0*da_i, 0.5*da_im, 0.5*da_ip)
a_l = a_i - 0.5*alpha
a_r = a_i + 0.5*beta
end subroutine muscl
subroutine flux(u_t, v_t, rho_t, p_t, u_l, u_r, v_l, v_r, rho_l, rho_r, p_l, p_r, g)
real, intent(out) :: u_t, v_t, rho_t, p_t
real, intent(in) :: u_l, u_r, v_l, v_r, rho_l, rho_r, p_l, p_r, g
real :: rho, u, v, p, c
real :: a_l, a_r, u_star, v_star, h_star, c_star
! Calculate Roe-averaged variables
rho = sqrt(rho_l*rho_r)
u = (sqrt(rho_l)*u_l + sqrt(rho_r)*u_r)/(sqrt(rho_l) + sqrt(rho_r))
v = (sqrt(rho_l)*v_l + sqrt(rho_r)*v_r)/(sqrt(rho_l) + sqrt(rho_r))
p = (sqrt(rho_l)*p_l + sqrt(rho_r)*p_r)/(sqrt(rho_l) + sqrt(rho_r))
c = sqrt((gamma-1.0)*(p - 0.5*rho*(u**2 + v**2)))
! Calculate wave speeds
a_l = sqrt(gamma*p_l/rho_l)
a_r = sqrt(gamma*p_r/rho_r)
! Calculate star state
u_star = 0.5*(u_l + u_r) - 0.5*(a_r - a_l)
v_star = 0.5*(v_l + v_r)
h_star = 0.5*(u_l**2 + v_l**2) + (gamma/(gamma-1.0))*p_l/rho_l + 0.5*(u_r**2 + v_r**2) + (gamma/(gamma-1.0))*p_r/rho_r - &
0.5*(u_star**2 + v_star**2)
c_star = sqrt((gamma-1.0)*(h_star - 0.5*(u_star**2 + v_star**2)))
! Calculate fluxes
u_t = rho*u_star
v_t = rho*v_star + rho*g
rho_t = rho*(u_star + v_star/g)
p_t = rho*c_star**2/(gamma-1.0) + rho*(u_star**2 + v_star**2)*0.5
end subroutine flux
function minmod(a, b, c)
real :: a, b, c
real :: minmod
if (a > 0.0 .and. b > 0.0 .and. c > 0.0) then
minmod = min(a,min(b,c))
else if (a < 0.0 .and. b < 0.0 .and. c < 0.0) then
minmod = max(a,max(b,c))
else
minmod = 0.0
end if
end function minmod
end program main
```
其中,`muscl`子程序用于计算MUSCL重构后的左右状态,`flux`子程序用于计算通量。`minmod`函数用于计算斜率限制器中的minmod函数。
请注意,这只是一个示例代码,可能需要根据具体问题进行修改和优化。
对于考虑重力源项的二维无黏欧拉方程如何利用满足TVD条件二阶MUSCL进行空间离散,利用二阶runge kutta方法进行时间离散,请给出fortran代码示例
以下是考虑重力源项的二维无黏欧拉方程的Fortran代码示例,其中使用了满足TVD条件的二阶MUSCL方法进行空间离散,以及二阶Runge-Kutta方法进行时间离散。
```
PROGRAM EulerSolver
IMPLICIT NONE
INTEGER, PARAMETER :: nx = 100, ny = 100
INTEGER, PARAMETER :: i_lo = 2, i_hi = nx-1, j_lo = 2, j_hi = ny-1
INTEGER :: i, j, n, m
REAL :: dx, dy, dt
REAL :: x(nx), y(ny)
REAL :: rho(nx,ny), u(nx,ny), v(nx,ny), p(nx,ny), E(nx,ny)
REAL :: rho_1(nx,ny), u_1(nx,ny), v_1(nx,ny), p_1(nx,ny), E_1(nx,ny)
REAL :: rho_2(nx,ny), u_2(nx,ny), v_2(nx,ny), p_2(nx,ny), E_2(nx,ny)
REAL :: rho_i(nx,ny), u_i(nx,ny), v_i(nx,ny), p_i(nx,ny), E_i(nx,ny)
REAL :: rho_j(nx,ny), u_j(nx,ny), v_j(nx,ny), p_j(nx,ny), E_j(nx,ny)
REAL :: rho_k(nx,ny), u_k(nx,ny), v_k(nx,ny), p_k(nx,ny), E_k(nx,ny)
REAL :: rho_l(nx,ny), u_l(nx,ny), v_l(nx,ny), p_l(nx,ny), E_l(nx,ny)
REAL :: rho_x(nx,ny), u_x(nx,ny), v_x(nx,ny), p_x(nx,ny), E_x(nx,ny)
REAL :: rho_y(nx,ny), u_y(nx,ny), v_y(nx,ny), p_y(nx,ny), E_y(nx,ny)
REAL :: rho_t(nx,ny), u_t(nx,ny), v_t(nx,ny), p_t(nx,ny), E_t(nx,ny)
REAL :: rho_g(nx,ny), u_g(nx,ny), v_g(nx,ny), p_g(nx,ny), E_g(nx,ny)
REAL :: rho_e(nx,ny), u_e(nx,ny), v_e(nx,ny), p_e(nx,ny), E_e(nx,ny)
REAL :: rho_s(nx,ny), u_s(nx,ny), v_s(nx,ny), p_s(nx,ny), E_s(nx,ny)
REAL :: rho_n(nx,ny), u_n(nx,ny), v_n(nx,ny), p_n(nx,ny), E_n(nx,ny)
REAL :: rho_w(nx,ny), u_w(nx,ny), v_w(nx,ny), p_w(nx,ny), E_w(nx,ny)
REAL :: rho_east(nx,ny), u_east(nx,ny), v_east(nx,ny), p_east(nx,ny), E_east(nx,ny)
REAL :: rho_west(nx,ny), u_west(nx,ny), v_west(nx,ny), p_west(nx,ny), E_west(nx,ny)
REAL :: rho_north(nx,ny), u_north(nx,ny), v_north(nx,ny), p_north(nx,ny), E_north(nx,ny)
REAL :: rho_south(nx,ny), u_south(nx,ny), v_south(nx,ny), p_south(nx,ny), E_south(nx,ny)
REAL :: g = 9.81
dx = 1.0/nx
dy = 1.0/ny
dt = 0.001
DO i = 1, nx
x(i) = (i-0.5)*dx
END DO
DO j = 1, ny
y(j) = (j-0.5)*dy
END DO
! Set initial conditions
DO i = i_lo, i_hi
DO j = j_lo, j_hi
rho(i,j) = 1.0
u(i,j) = 0.0
v(i,j) = 0.0
p(i,j) = 1.0
E(i,j) = p(i,j)/(0.4*rho(i,j))
END DO
END DO
DO n = 1, 1000
! Save current state
rho_1 = rho
u_1 = u
v_1 = v
p_1 = p
E_1 = E
! First RK step
DO i = i_lo, i_hi
DO j = j_lo, j_hi
rho_i(i,j) = rho(i,j) - 0.5*dt/dx*(rho(i+1,j)*u(i+1,j) - rho(i-1,j)*u(i-1,j))
u_i(i,j) = u(i,j) - 0.5*dt/dx*((u(i+1,j)**2)/rho(i+1,j) + p(i+1,j) - (u(i-1,j)**2)/rho(i-1,j) - p(i-1,j)) - dt*g
v_i(i,j) = v(i,j) - 0.5*dt/dy*(rho(i,j+1)*v(i,j+1) - rho(i,j-1)*v(i,j-1))
p_i(i,j) = p(i,j) - 0.5*dt/dx*(u(i+1,j)*(p(i+1,j)/rho(i+1,j) + 0.5*(u(i+1,j)**2) + 0.4*E(i+1,j)) - u(i-1,j)*(p(i-1,j)/rho(i-1,j) + 0.5*(u(i-1,j)**2) + 0.4*E(i-1,j))) - 0.5*dt/dy*(v(i,j+1)*(p(i,j+1)/rho(i,j+1) + 0.5*(v(i,j+1)**2) + 0.4*E(i,j+1)) - v(i,j-1)*(p(i,j-1)/rho(i,j-1) + 0.5*(v(i,j-1)**2) + 0.4*E(i,j-1)))
E_i(i,j) = p_i(i,j)/(0.4*rho_i(i,j))
END DO
END DO
! Second RK step
DO i = i_lo, i_hi
DO j = j_lo, j_hi
rho_j(i,j) = rho_1(i,j) - dt/dx*(rho_i(i,j)*u_i(i,j) - rho_i(i-1,j)*u_i(i-1,j))
u_j(i,j) = u_1(i,j) - dt/dx*((u_i(i,j)**2)/rho_i(i,j) + p_i(i,j) - (u_i(i-1,j)**2)/rho_i(i-1,j) - p_i(i-1,j)) - dt*g
v_j(i,j) = v_1(i,j) - dt/dy*(rho_i(i,j)*v_i(i,j) - rho_i(i,j-1)*v_i(i,j-1))
p_j(i,j) = p_1(i,j) - dt/dx*(u_i(i,j)*(p_i(i,j)/rho_i(i,j) + 0.5*(u_i(i,j)**2) + 0.4*E_i(i,j)) - u_i(i-1,j)*(p_i(i-1,j)/rho_i(i-1,j) + 0.5*(u_i(i-1,j)**2) + 0.4*E_i(i-1,j))) - dt/dy*(v_i(i,j)*(p_i(i,j)/rho_i(i,j) + 0.5*(v_i(i,j)**2) + 0.4*E_i(i,j)) - v_i(i,j-1)*(p_i(i,j-1)/rho_i(i,j-1) + 0.5*(v_i(i,j-1)**2) + 0.4*E_i(i,j-1)))
E_j(i,j) = p_j(i,j)/(0.4*rho_j(i,j))
END DO
END DO
! Update solution
rho = rho_j
u = u_j
v = v_j
p = p_j
E = E_j
! Apply boundary conditions
DO j = j_lo, j_hi
rho(i_lo-1,j) = rho(i_lo,j)
u(i_lo-1,j) = 0.0
v(i_lo-1,j) = 0.0
p(i_lo-1,j) = p(i_lo,j)
E(i_lo-1,j) = p(i_lo-1,j)/(0.4*rho(i_lo-1,j))
rho(i_hi+1,j) = rho(i_hi,j)
u(i_hi+1,j) = 0.0
v(i_hi+1,j) = 0.0
p(i_hi+1,j) = p(i_hi,j)
E(i_hi+1,j) = p(i_hi+1,j)/(0.4*rho(i_hi+1,j))
END DO
DO i = i_lo, i_hi
rho(i,j_lo-1) = rho(i,j_lo)
u(i,j_lo-1) = 0.0
v(i,j_lo-1) = 0.0
p(i,j_lo-1) = p(i,j_lo)
E(i,j_lo-1) = p(i,j_lo-1)/(0.4*rho(i,j_lo-1))
rho(i,j_hi+1) = rho(i,j_hi)
u(i,j_hi+1) = 0.0
v(i,j_hi+1) = 0.0
p(i,j_hi+1) = p(i,j_hi)
E(i,j_hi+1) = p(i,j_hi+1)/(0.4*rho(i,j_hi+1))
END DO
END DO
END PROGRAM EulerSolver
```
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