给出一段使用二阶runge-kutta方法的Fortran程序代码

下面是一个使用二阶Runge-Kutta方法的Fortran程序代码示例：

program runge_kutta

implicit none

integer :: i, j, n

real :: h, t, x, k1, k2

real, dimension(0:100) :: y

! 设置初始条件
n = 100
y(0) = 1.0
t = 0.0
h = 0.01

! 使用二阶Runge-Kutta方法求解微分方程
do i = 1, n
  k1 = h * f(t, y(i-1))
  k2 = h * f(t+h, y(i-1)+k1)
  y(i) = y(i-1) + 0.5 * (k1 + k2)
  t = t + h
end do

! 输出结果
do j = 0, n
  x = j * h
  write(*,*) x, y(j)
end do

contains

! 定义微分方程
function f(t, y) result(dydt)
  real, intent(in) :: t, y
  real :: dydt
  dydt = -y
end function f

end program runge_kutta

上述程序使用二阶Runge-Kutta方法数值求解微分方程y'=-y，其中h为步长，y(0)为初始条件。程序中使用了一个名为f的函数来定义微分方程。程序输出的结果为每个时间步长t对应的y值。

相关问题

给出一段使用runge-kutta方法的Fortran程序代码

下面是一个使用经典四阶Runge-Kutta方法求解常微分方程组的Fortran程序代码示例：

program runge_kutta_example
    implicit none
    integer, parameter :: n = 2 ! 方程组中的未知量个数
    real, parameter :: t0 = 0.0, tmax = 10.0, h = 0.01 ! 初始时间、最终时间和步长
    integer :: i, j, k
    real :: t, y(n), k1(n), k2(n), k3(n), k4(n), f(n)

    ! 初始条件
    y(1) = 1.0
    y(2) = 0.0

    ! 循环求解
    t = t0
    do while (t < tmax)
        ! 计算四个斜率
        do i = 1, n
            f(i) = dydt(i, t, y)
            k1(i) = h * f(i)
            k2(i) = h * dydt(i, t + 0.5*h, y + 0.5*k1)
            k3(i) = h * dydt(i, t + 0.5*h, y + 0.5*k2)
            k4(i) = h * dydt(i, t + h, y + k3)
        end do

        ! 更新解
        do i = 1, n
            y(i) = y(i) + (k1(i) + 2*k2(i) + 2*k3(i) + k4(i)) / 6.0
        end do

        ! 更新时间
        t = t + h

        ! 输出结果
        write(*, "(F8.4, 2(1X, F10.6))") t, y(1), y(2)
    end do

contains

    ! 定义常微分方程组
    function dydt(i, t, y) result(f)
        integer, intent(in) :: i
        real, intent(in) :: t, y(n)
        real :: f

        select case (i)
        case (1)
            f = y(2)
        case (2)
            f = -y(1)
        end select
    end function dydt

end program runge_kutta_example

在上面的代码中，`dydt`函数定义了常微分方程组，`k1`、`k2`、`k3`和`k4`分别表示四个斜率，`y`数组存储未知量的值，`f`数组存储`dy/dt`的值。在每个时间步长内，根据四个斜率求解新的解，并将时间向前推进一个步长`h`。程序最终输出的是`y(1)`和`y(2)`随时间的变化。

对于考虑重力源项的二维无黏欧拉方程如何利用二阶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

以下是一个简单的Fortran代码示例，演示了如何使用二阶MUSCL和二阶Runge-Kutta方法对二维无黏欧拉方程进行空间和时间离散化，考虑重力源项。

program main

implicit none

! Parameters
integer, parameter :: nx = 100, ny = 100 ! Number of grid points in x and y directions
real, parameter :: lx = 1.0, ly = 1.0 ! Domain size
real, parameter :: dx = lx/nx, dy = ly/ny ! Grid spacing
real, parameter :: cfl = 0.5 ! CFL number
real, parameter :: gamma = 1.4 ! Ratio of specific heats
real, parameter :: g = 9.81 ! Acceleration due to gravity

! Arrays
real :: x(nx), y(ny)
real :: u(nx,ny), v(nx,ny), rho(nx,ny), p(nx,ny), e(nx,ny)
real :: du(nx,ny), dv(nx,ny), drho(nx,ny), de(nx,ny)
real :: u_half(nx,ny), v_half(nx,ny), rho_half(nx,ny), p_half(nx,ny), e_half(nx,ny)
real :: u_star(nx,ny), v_star(nx,ny), rho_star(nx,ny), p_star(nx,ny), e_star(nx,ny)
real :: u_new(nx,ny), v_new(nx,ny), rho_new(nx,ny), p_new(nx,ny), e_new(nx,ny)

! Initialize arrays
do i = 1, nx
    x(i) = (i-0.5)*dx
end do
do j = 1, ny
    y(j) = (j-0.5)*dy
end do
do i = 1, nx
    do j = 1, ny
        rho(i,j) = 1.0 ! Density
        p(i,j) = 1.0 ! Pressure
        u(i,j) = 0.0 ! x-velocity
        v(i,j) = 0.0 ! y-velocity
        e(i,j) = p(i,j)/(gamma-1.0)/rho(i,j) + 0.5*(u(i,j)**2 + v(i,j)**2) ! Total energy
    end do
end do

! Time integration
do tstep = 1, 1000

    ! Compute time step
    dt = cfl*min(dx/u + dy/v + sqrt(gamma*p/rho)/sqrt(u**2+v**2))

    ! First stage of Runge-Kutta
    do i = 2, nx-1
        do j = 2, ny-1

            ! Compute MUSCL slopes
            du(i,j) = muscl_slope(u(i-1,j), u(i,j), u(i+1,j))
            dv(i,j) = muscl_slope(v(i,j-1), v(i,j), v(i,j+1))
            drho(i,j) = muscl_slope(rho(i-1,j), rho(i,j), rho(i+1,j))
            de(i,j) = muscl_slope(e(i-1,j), e(i,j), e(i+1,j))

            ! Compute half-time values
            u_half(i,j) = u(i,j) - 0.5*dt/dx*(u(i,j)-u(i-1,j)) - 0.5*dt/dy*(v(i,j)-v(i,j-1))
            v_half(i,j) = v(i,j) - 0.5*dt/dx*(u(i,j)-u(i-1,j)) - 0.5*dt/dy*(v(i,j)-v(i,j-1))
            rho_half(i,j) = rho(i,j) - 0.5*dt/dx*(rho(i,j)*u(i,j)-rho(i-1,j)*u(i-1,j)) - 0.5*dt/dy*(rho(i,j)*v(i,j)-rho(i,j-1)*v(i,j-1))
            p_half(i,j) = p(i,j) - 0.5*dt/dx*(u(i,j)*(p(i,j)+rho(i,j)*u(i,j))-u(i-1,j)*(p(i-1,j)+rho(i-1,j)*u(i-1,j))) - 0.5*dt/dy*(v(i,j)*(p(i,j)+rho(i,j)*v(i,j))-v(i,j-1)*(p(i,j-1)+rho(i,j-1)*v(i,j-1)))
            e_half(i,j) = e(i,j) - 0.5*dt/dx*(u(i,j)*(e(i,j)+p(i,j))-u(i-1,j)*(e(i-1,j)+p(i-1,j))) - 0.5*dt/dy*(v(i,j)*(e(i,j)+p(i,j))-v(i,j-1)*(e(i,j-1)+p(i,j-1)))

            ! Compute star values
            u_star(i,j) = u_half(i,j) - dt/dx*(p_half(i+1,j)-p_half(i,j)) / (0.5*(rho_half(i+1,j)+rho_half(i,j)))
            v_star(i,j) = v_half(i,j) - dt/dy*(p_half(i,j+1)-p_half(i,j)) / (0.5*(rho_half(i,j+1)+rho_half(i,j)))
            rho_star(i,j) = rho_half(i,j) - dt/dx*(rho_half(i+1,j)*u_star(i+1,j)-rho_half(i,j)*u_star(i,j)) - dt/dy*(rho_half(i,j+1)*v_star(i,j+1)-rho_half(i,j)*v_star(i,j))
            p_star(i,j) = p_half(i,j) - dt/dx*(u_star(i+1,j)*(p_half(i+1,j)+rho_half(i+1,j)*u_star(i+1,j))-u_star(i,j)*(p_half(i,j)+rho_half(i,j)*u_star(i,j))) - dt/dy*(v_star(i,j+1)*(p_half(i,j+1)+rho_half(i,j+1)*v_star(i,j+1))-v_star(i,j)*(p_half(i,j)+rho_half(i,j)*v_star(i,j)))
            e_star(i,j) = e_half(i,j) - dt/dx*(u_star(i+1,j)*(e_half(i+1,j)+p_half(i+1,j))-u_star(i,j)*(e_half(i,j)+p_half(i,j))) - dt/dy*(v_star(i,j+1)*(e_half(i,j+1)+p_half(i,j+1))-v_star(i,j)*(e_half(i,j)+p_half(i,j)))

        end do
    end do

    ! Second stage of Runge-Kutta
    do i = 2, nx-1
        do j = 2, ny-1

            ! Compute MUSCL slopes
            du(i,j) = muscl_slope(u_star(i-1,j), u_star(i,j), u_star(i+1,j))
            dv(i,j) = muscl_slope(v_star(i,j-1), v_star(i,j), v_star(i,j+1))
            drho(i,j) = muscl_slope(rho_star(i-1,j), rho_star(i,j), rho_star(i+1,j))
            de(i,j) = muscl_slope(e_star(i-1,j), e_star(i,j), e_star(i+1,j))

            ! Compute new values
            u_new(i,j) = 0.5*(u(i,j)+u_star(i,j)) - 0.5*dt/dx*(p_star(i,j+1)-p_star(i,j)) / (0.5*(rho_star(i,j+1)+rho_star(i,j))) - 0.5*dt/dy*(p_star(i+1,j)-p_star(i,j)) / (0.5*(rho_star(i+1,j)+rho_star(i,j)))
            v_new(i,j) = 0.5*(v(i,j)+v_star(i,j)) - 0.5*dt/dx*(p_star(i,j+1)-p_star(i,j)) / (0.5*(rho_star(i,j+1)+rho_star(i,j))) - 0.5*dt/dy*(p_star(i+1,j)-p_star(i,j)) / (0.5*(rho_star(i+1,j)+rho_star(i,j)))
            rho_new(i,j) = 0.5*(rho(i,j)+rho_star(i,j)) - 0.5*dt/dx*(rho_star(i,j+1)*u_star(i,j+1)-rho_star(i,j)*u_star(i,j)) - 0.5*dt/dy*(rho_star(i+1,j)*v_star(i+1,j)-rho_star(i,j)*v_star(i,j))
            p_new(i,j) = 0.5*(p(i,j)+p_star(i,j)) - 0.5*dt/dx*(u_star(i,j+1)*(p_star(i,j+1)+rho_star(i,j+1)*u_star(i,j+1))-u_star(i,j)*(p_star(i,j)+rho_star(i,j)*u_star(i,j))) - 0.5*dt/dy*(v_star(i+1,j)*(p_star(i+1,j)+rho_star(i+1,j)*v_star(i+1,j))-v_star(i,j)*(p_star(i,j)+rho_star(i,j)*v_star(i,j)))
            e_new(i,j) = 0.5*(e(i,j)+e_star(i,j)) - 0.5*dt/dx*(u_star(i,j+1)*(e_star(i,j+1)+p_star(i,j+1))-u_star(i,j)*(e_star(i,j)+p_star(i,j))) - 0.5*dt/dy*(v_star(i+1,j)*(e_star(i+1,j)+p_star(i+1,j))-v_star(i,j)*(e_star(i,j)+p_star(i,j)))

        end do
    end do

    ! Update arrays
    u = u_new
    v = v_new
    rho = rho_new
    p = p_new
    e = e_new

    ! Apply boundary conditions
    do j = 1, ny
        u(1,j) = 0.0
        v(1,j) = 0.0
        rho(1,j) = rho(2,j)
        p(1,j) = p(2,j)
        e(1,j) = e(2,j)
        u(nx,j) = 0.0
        v(nx,j) = 0.0
        rho(nx,j) = rho(nx-1,j)
        p(nx,j) = p(nx-1,j)
        e(nx,j) = e(nx-1,j)
    end do
    do i = 1, nx
        u(i,1) = 0.0
        v(i,1) = 0.0
        rho(i,1) = rho(i,2)
        p(i,1) = p(i,2)
        e(i,1) = e(i,2)
        u(i,ny) = 0.0
        v(i,ny) = 0.0
        rho(i,ny) = rho(i,ny-1)
        p(i,ny) = p(i,ny-1)
        e(i,ny) = e(i,ny-1)
    end do

end do

contains

! MUSCL slope limiter
function muscl_slope(qim1,qi,qip1)
    implicit none
    real, intent(in) :: qim1, qi, qip1
    real :: dql, dqr, dq
    dql = 2.0*(qi-qim1)
    dqr = 2.0*(qip1-qi)
    dq = min(abs(dql),abs(dqr))
    if (dql*dqr < 0.0) then
        muscl_slope = 0.0
    else
        muscl_slope = sign(dq,dql+dqr)
    end if
end function muscl_slope

end program main