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，利用满足TVD条件二阶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，用于模拟考虑重力源项的二维无黏欧拉方程。该代码使用二阶MUSCL空间离散和二阶Runge-Kutta时间离散，并使用基于TVD条件的限制器来确保数值稳定性。

PROGRAM EulerEquations

  IMPLICIT NONE

  ! Grid parameters
  INTEGER, PARAMETER :: nx = 100, ny = 50
  REAL :: x_min = 0.0, x_max = 1.0, y_min = 0.0, y_max = 0.5
  REAL :: dx = (x_max - x_min) / REAL(nx - 1)
  REAL :: dy = (y_max - y_min) / REAL(ny - 1)

  ! Physical parameters
  REAL, PARAMETER :: gamma = 1.4
  REAL, PARAMETER :: pi = 3.141592653589793
  REAL, PARAMETER :: g = 9.8

  ! Time parameters
  REAL :: t_start = 0.0, t_end = 1.0
  REAL :: dt = 0.001
  INTEGER :: n_steps

  ! Solution arrays
  REAL :: rho(nx, ny), u(nx, ny), v(nx, ny), p(nx, ny)
  REAL :: rho_t(nx, ny), u_t(nx, ny), v_t(nx, ny), p_t(nx, ny)
  REAL :: rho_l(nx, ny), u_l(nx, ny), v_l(nx, ny), p_l(nx, ny)
  REAL :: rho_r(nx, ny), u_r(nx, ny), v_r(nx, ny), p_r(nx, ny)
  REAL :: rho_b(nx, ny), u_b(nx, ny), v_b(nx, ny), p_b(nx, ny)
  REAL :: rho_t_b(nx, ny), u_t_b(nx, ny), v_t_b(nx, ny), p_t_b(nx, ny)
  REAL :: rho_l_r(nx, ny), u_l_r(nx, ny), v_l_r(nx, ny), p_l_r(nx, ny)

  ! Solution initialization
  DO j = 1, ny
    DO i = 1, nx
      x = x_min + REAL(i - 1) * dx
      y = y_min + REAL(j - 1) * dy
      rho(i, j) = 1.0 + 0.2 * SIN(pi * x) * SIN(pi * y)
      u(i, j) = 0.5
      v(i, j) = 0.5
      p(i, j) = 1.0
    END DO
  END DO

  ! Time integration loop
  n_steps = INT((t_end - t_start) / dt)
  DO step = 1, n_steps

    ! Time step 1
    rho_t = rho - dt * (rho * u_x + rho * v_y)
    u_t = u - dt * ((rho * u) * u_x + (rho * v) * u_y + p_x)
    v_t = v - dt * ((rho * u) * v_x + (rho * v) * v_y + p_y)
    p_t = p - dt * (gamma * p * (u_x + v_y) + rho * g * SIN(pi * y))

    ! Apply TVD limiter
    CALL TVD_Limiter(rho_t, u_t, v_t, p_t)

    ! Time step 2
    rho_l = 0.5 * (rho + rho_t - dt * (rho_t * u_x + rho_t * v_y))
    u_l = 0.5 * (u + u_t - dt * ((rho_t * u_t) * u_x + (rho_t * v_t) * u_y + p_t_x))
    v_l = 0.5 * (v + v_t - dt * ((rho_t * u_t) * v_x + (rho_t * v_t) * v_y + p_t_y))
    p_l = 0.5 * (p + p_t - dt * (gamma * p_t * (u_t_x + v_t_y) + rho_t * g * SIN(pi * y)))

    rho_r = 0.5 * (rho_t + rho - dt * (rho * u_x + rho * v_y))
    u_r = 0.5 * (u_t + u - dt * ((rho * u) * u_x + (rho * v) * u_y + p_x))
    v_r = 0.5 * (v_t + v - dt * ((rho * u) * v_x + (rho * v) * v_y + p_y))
    p_r = 0.5 * (p_t + p - dt * (gamma * p * (u_x + v_y) + rho * g * SIN(pi * y)))

    rho_b = 0.5 * (rho + rho_t - dt * (rho_t * u_x + rho_t * v_y))
    u_b = 0.5 * (u + u_t - dt * ((rho_t * u_t) * u_x + (rho_t * v_t) * u_y + p_t_x))
    v_b = 0.5 * (v + v_t - dt * ((rho_t * u_t) * v_x + (rho_t * v_t) * v_y + p_t_y))
    p_b = 0.5 * (p + p_t - dt * (gamma * p_t * (u_t_x + v_t_y) + rho_t * g * SIN(pi * y)))

    rho_t_b = 0.5 * (rho_t + rho_b - dt * (rho_b * u_x + rho_b * v_y))
    u_t_b = 0.5 * (u_t + u_b - dt * ((rho_b * u_b) * u_x + (rho_b * v_b) * u_y + p_b_x))
    v_t_b = 0.5 * (v_t + v_b - dt * ((rho_b * u_b) * v_x + (rho_b * v_b) * v_y + p_b_y))
    p_t_b = 0.5 * (p_t + p_b - dt * (gamma * p_b * (u_b_x + v_b_y) + rho_b * g * SIN(pi * y)))

    rho_l_r = 0.5 * (rho_r + rho_l - dt * (rho_l * u_x + rho_r * u_x))
    u_l_r = 0.5 * (u_r + u_l - dt * ((rho_l * u_l) * u_x + (rho_r * u_r) * u_x + p_l_x))
    v_l_r = 0.5 * (v_r + v_l - dt * ((rho_l * u_l) * v_x + (rho_r * u_r) * v_x + p_l_y))
    p_l_r = 0.5 * (p_r + p_l - dt * (gamma * p_l * (u_l_x + v_l_y) + rho_l * g * SIN(pi * y)))

    ! Apply TVD limiter
    CALL TVD_Limiter(rho_l_r, u_l_r, v_l_r, p_l_r)

    ! Update solution
    rho = rho - dt * (rho_l_r * u_x + rho_t_b * v_y)
    u = u - dt * ((rho_l_r * u_l_r + p_l_r) * u_x + (rho_t_b * u_t_b + p_t_b) * v_y)
    v = v - dt * ((rho_l_r * v_l_r + p_l_r) * u_x + (rho_t_b * v_t_b + p_t_b) * v_y)
    p = p - dt * (gamma * p_l_r * (u_l_r_x + v_l_r_y))

  END DO

CONTAINS

  ! TVD limiter subroutine
  SUBROUTINE TVD_Limiter(rho, u, v, p)

    IMPLICIT NONE

    REAL, INTENT(INOUT) :: rho(:,:), u(:,:), v(:,:), p(:,:)
    REAL :: r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12
    INTEGER :: i, j

    DO j = 2, ny - 1
      DO i = 2, nx - 1

        ! Calculate slopes
        r1 = (rho(i, j) - rho(i - 1, j)) / dx
        r2 = (rho(i + 1, j) - rho(i, j)) / dx
        r3 = (rho(i, j) - rho(i, j - 1)) / dy
        r4 = (rho(i, j + 1) - rho(i, j)) / dy
        r5 = (u(i, j) - u(i - 1, j)) / dx
        r6 = (u(i + 1, j) - u(i, j)) / dx
        r7 = (u(i, j) - u(i, j - 1)) / dy
        r8 = (u(i, j + 1) - u(i, j)) / dy
        r9 = (v(i, j) - v(i - 1, j)) / dx
        r10 = (v(i + 1, j) - v(i, j)) / dx
        r11 = (v(i, j) - v(i, j - 1)) / dy
        r12 = (v(i, j + 1) - v(i, j)) / dy

        ! Calculate TVD coefficients
        alpha_rho = MINMOD(r1 / (EPSILON + rho(i, j)), r2 / (EPSILON + rho(i, j)))
        beta_rho = MINMOD(r3 / (EPSILON + rho(i, j)), r4 / (EPSILON + rho(i, j)))
        alpha_u = MINMOD(r5 / (EPSILON + u(i, j)), r6 / (EPSILON + u(i, j)))
        beta_u = MINMOD(r7 / (EPSILON + u(i, j)), r8 / (EPSILON + u(i, j)))
        alpha_v = MINMOD(r9 / (EPSILON + v(i, j)), r10 / (EPSILON + v(i, j)))
        beta_v = MINMOD(r11 / (EPSILON + v(i, j)), r12 / (EPSILON + v(i, j)))

        ! Apply TVD limiter
        rho(i, j) = rho(i, j) + 0.5 * alpha_rho * dx * rho(i, j) * SIGN(alpha_rho)
        u(i, j) = u(i, j) + 0.5 * alpha_u * dx * u(i, j) * SIGN(alpha_u)
        v(i, j) = v(i, j) + 0.5 * alpha_v * dx * v(i, j) * SIGN(alpha_v)
        p(i, j) = p(i, j) + 0.5 * (alpha_rho * beta_u + alpha_u * beta_rho) * dx * p(i, j) * SIGN(alpha_rho * beta_u + alpha_u * beta_rho)

      END DO
    END DO

  END SUBROUTINE TVD_Limiter

  ! MINMOD function
  FUNCTION MINMOD(a, b)
    IMPLICIT NONE
    REAL, INTENT(IN) :: a, b
    REAL :: MINMOD
    MINMOD = 0.5 * (SIGN(a) + SIGN(b)) * MIN(ABS(a), ABS(b))
  END FUNCTION MINMOD

END PROGRAM EulerEquations

请注意，这只是一个简单的示例，并且可能需要进行修改以满足您的特定需求。此外，该代码中的一些细节可能需要进一步解释，特别是涉及TVD限制器的部分。