euler_1D_caf_nf Derived Type

type :: euler_1D_caf_nf
  !< Euler 1D (CAF enabled) PDEs system field.
  !<
  !< @note This object does not use FOODIE.
  !<
  !<### 1D Euler PDEs system
  !< The 1D Euler PDEs system considered is a non linear, hyperbolic (inviscid) system of conservation laws for compressible gas
  !< dynamics, that reads as
  !<$$
  !<\begin{matrix}
  !<U_t = R(U)  \Leftrightarrow U_t = F(U)_x \\
  !<U = \begin{bmatrix}
  !<\rho \\
  !<\rho u \\
  !<\rho E
  !<\end{bmatrix}\;\;\;
  !<F(U) = \begin{bmatrix}
  !<\rho u \\
  !<\rho u^2 + p \\
  !<\rho u H
  !<\end{bmatrix}
  !<\end{matrix}
  !<$$
  !< where \(\rho\) is the density, \(u\) is the velocity, \(p\) the pressure, \(E\) the total internal specific energy and \(H\)
  !< the total specific enthalpy. The PDEs system must completed with the proper initial and boundary conditions. Moreover, an ideal
  !< (thermally and calorically perfect) gas is considered
  !<$$
  !<\begin{matrix}
  !<R = c_p - c_v \\
  !<\gamma = \frac{c_p}{c_v}\\
  !<e = c_v T \\
  !<h = c_p T
  !<\end{matrix}
  !<$$
  !< where *R* is the gas constant, \(c_p\,c_v\) are the specific heats at constant pressure and volume (respectively), *e* is the
  !< internal energy, *h* is the internal enthalpy and *T* is the temperature. The following addition equations of state hold:
  !<$$
  !<\begin{matrix}
  !<T = \frac{p}{\rho R} \\
  !<E = \rho e + \frac{1}{2} \rho u^2 \\
  !<H = \rho h + \frac{1}{2} \rho u^2 \\
  !<a = \sqrt{\frac{\gamma p}{\rho}}
  !<\end{matrix}
  !<$$
  !<
  !<### Multi-fluid Euler PDEs system
  !< An extension of the above Euler system is considered allowing the modelling of a multi-fluid mixture of different gas (with
  !< different physical characteristics). The well known Standard Thermodynamic Model is used to model the gas mixture replacing the
  !< density with the density fraction of each specie composing the mixture. This led to the following system:
  !<$$
  !<\begin{matrix}
  !<U_t = R(U)  \Leftrightarrow U_t = F(U)_x \\
  !<U = \begin{bmatrix}
  !<\rho_s \\
  !<\rho u \\
  !<\rho E
  !<\end{bmatrix}\;\;\;
  !<F(U) = \begin{bmatrix}
  !<\rho_s u \\
  !<\rho u^2 + p \\
  !<\rho u H
  !<\end{bmatrix}\;\;\; for\; s=1,2,...N_s \\
  !<\rho = \sum_{s=1}^{N_s}\rho_s \\
  !<c_p = \sum_{s=1}^{N_S} \frac{\rho_s}{\rho} c_{p,s} \quad  c_v = \sum_{s=1}^{N_S} \frac{\rho_s}{\rho} c_{v,s}
  !<\end{matrix}
  !<$$
  !< where \(N_s\) is the number of initial species composing the gas mixture.
  !<
  !<#### Numerical grid organization
  !< The finite volume, Godunov's like approach is employed. The conservative variables (and the primitive ones) are co-located at
  !< the cell center. The cell and (inter)faces numeration is as follow.
  !<```
  !<                cell            (inter)faces
  !<                 |                   |
  !<                 v                   v
  !<     |-------|-------|-.....-|-------|-------|-------|-------|-.....-|-------|-------|-------|-.....-|-------|-------|
  !<     | 1-Ng  | 2-Ng  | ..... |  -1   |   0   |   1   |  2    | ..... |  Ni   | Ni+1  | Ni+1  | ..... |Ni+Ng-1| Ni+Ng |
  !<     |-------|-------|-.....-|-------|-------|-------|-------|-.....-|-------|-------|-------|-.....-|-------|-------|
  !<    0-Ng                             -1      0       1       2      Ni-1     Ni                                    Ni+Ng
  !<```
  !< Where *Ni* are the finite volumes (cells) used for discretizing the domain and *Ng* are the ghost cells used for imposing the
  !< left and right boundary conditions (for a total of *2Ng* cells).
  !<
  !<#### Primitive variables organization
  !< Primitive variables are organized as an array of reals which the first index means:
  !<
  !< + 1    : density of species 1    (r1)
  !< + 2    : density of species 2    (r2)
  !< + ...  :
  !< + s    : density of species s-th (rs)
  !< + ...  :
  !< + Ns   : density of species Ns   (rNs)
  !< + Ns+1 : velocity                (u)
  !< + Ns+2 : pressure                (p)
  !< + Ns+3 : density                 (r=sum(rs))
  !< + Ns+4 : specific heats ratio    (g)
  !<
  !<#### Conservative variables organization
  !< Conservative variables are organized as an array (rank 2) of reals which the first index means:
  !<
  !< + 1    : mass conservation of species 1    (r1)
  !< + 2    : mass conservation of species 2    (r2)
  !< + ...  :
  !< + s    : mass conservation of species s-th (rs)
  !< + ...  :
  !< + Ns   : mass conservation of species Ns   (rNs)
  !< + Ns+1 : momentum conservation             (r*u)
  !< + Ns+2 : energy conservation               (r*E)
  private
  integer(I_P)                   :: ord=0     !< Space accuracy formal order.
  integer(I_P)                   :: Ni=0      !< Space dimension.
  integer(I_P)                   :: Ng=0      !< Number of ghost cells for boundary conditions handling.
  integer(I_P)                   :: Ns=0      !< Number of initial species.
  integer(I_P)                   :: Nc=0      !< Number of conservative variables, Ns+2.
  integer(I_P)                   :: Np=0      !< Number of primitive variables, Ns+4.
  real(R_P)                      :: Dx=0._R_P !< Space step.
  type(weno_interpolator_upwind) :: weno      !< WENO interpolator.
  real(R_P),    allocatable      :: U(:,:)    !< Integrand (state) variables, whole physical domain [1:Nc,1:Ni].
  real(R_P),    allocatable      :: cp0(:)    !< Specific heat cp of initial species [1:Ns].
  real(R_P),    allocatable      :: cv0(:)    !< Specific heat cv of initial species [1:Ns].
  character(:), allocatable      :: BC_L      !< Left boundary condition type.
  character(:), allocatable      :: BC_R      !< Right boundary condition type.
  integer(I_P)                   :: me=0      !< ID of this_image().
  integer(I_P)                   :: we=0      !< Number of CAF images used.
  contains
    ! auxiliary methods
    procedure, pass(self), public :: init                              !< Init field.
    procedure, pass(self), public :: destroy                           !< Destroy field.
    procedure, pass(self), public :: output                            !< Extract Euler field.
    procedure, pass(self), public :: dt => compute_dt                  !< Compute the current time step, by means of CFL condition.
    generic,               public :: assignment(=) => assign_integrand !< Overloading = assignament.
    ! private methods
    procedure, pass(self), private :: t => dEuler_dt                         !< Time derivative, residuals function.
    procedure, pass(lhs),  private :: assign_integrand => euler_assign_euler !< Euler = Euler.
    procedure, pass(self), private :: primitive2conservative                 !< Convert primitive variables to conservative ones.
    procedure, pass(self), private :: conservative2primitive                 !< Convert conservative variables to primitive ones.
    procedure, pass(self), private :: synchronize                            !< Synchronize CAF images.
    procedure, pass(self), private :: impose_boundary_conditions             !< Impose boundary conditions.
    procedure, pass(self), private :: reconstruct_interfaces_states          !< Reconstruct interfaces states.
    procedure, pass(self), private :: riemann_solver                         !< Solve the Riemann Problem at cell interfaces.
endtype euler_1D_caf_nf