CHASE

CFD-HPC enabled, Adaptive mesh, Simulation code for Euler equations.

CHASE solves the compressible inviscid Euler equations on block-structured adaptive meshes. It is built on the ADAM SDK and shares the same common/ + backend pattern as NASTO, but targets the Euler regime (no viscosity or heat diffusion). Only a CPU backend is provided.

Governing equations

The compressible Euler equations in conservation form:

\frac{\partial q}{\partial t} + \nabla \cdot F (q) = 0

with conservative variables $q = (ρ, ρ u, ρ v, ρ w, ρ E)^{⊤}$ and ideal-gas closure $p = ρ R T$ , $E = c_{v} T + \frac{1}{2} (u^{2} + v^{2} + w^{2})$ .

The Euler flux in the $x$ -direction is:

F_{x} = (\begin{matrix} ρ u \\ ρ u^{2} + p \\ ρ v u \\ ρ w u \\ ρ u H \end{matrix}), H = E + \frac{p}{ρ}

with analogous expressions for $F_{y}$ and $F_{z}$ .

Source layout

src/app/chase/
├── common/                              # Backend-independent modules
│   ├── adam_chase_common_object.F90     # Base object — ADAM SDK + CHASE handlers
│   ├── adam_chase_physics_object.F90    # Fluid physics — EOS, variable layout, flux functions
│   ├── adam_chase_parameters.F90        # Global constants and eigenvector matrices
│   ├── adam_chase_bc_object.F90         # Boundary conditions handler
│   ├── adam_chase_ic_object.F90         # Initial conditions handler
│   ├── adam_chase_io_object.F90         # IO handler (XH5F output, restart, residuals)
│   ├── adam_chase_time_object.F90       # Time integration parameters and state
│   ├── adam_chase_riemann_library.F90   # Riemann solver library (LLF, HLL)
│   └── adam_chase_common_library.F90    # Barrel re-export of all common modules
└── cpu/                                 # CPU backend
    ├── adam_chase_cpu.F90               # Entry point — parses CLI, calls simulate
    └── adam_chase_cpu_object.F90        # Backend object — working arrays, all methods

Variable layout

Conservative variables — `q(nv=5, ni, nj, nk, nb)`

Index	Symbol	Description
1	$ρ$	Density
2	$ρ u$	x-momentum
3	$ρ v$	y-momentum
4	$ρ w$	z-momentum
5	$ρ E$	Total energy

Auxiliary variables — `q_aux(nv_aux=11, ni, nj, nk, nb)`

Index	Symbol	Description
1	$ρ$	Density
2–4	$u, v, w$	Velocity components
5	$p$	Pressure
6	$T$	Temperature
7	$E$	Total specific energy
8	$H$	Total specific enthalpy
9	$a$	Speed of sound
10	$R_{f} = c_{p} - c_{v}$	Fluid constant
11	$γ = c_{p} / c_{v}$	Specific heats ratio

`adam_chase_common_object`

Base (non-extending) type that holds all ADAM SDK objects and CHASE handlers. Both backends compose the CPU object by extending this type.

Data members

Member	Type	Description
`mpih`	`mpih_object`	MPI handler
`adam`	`adam_object`	ADAM SDK — grid, tree, field, AMR, IO
`field`	`field_object` (pointer)	Field variable storage and ghost-cell maps
`grid`	`grid_object` (pointer)	Structured block grid
`amr`	`amr_object`	AMR marker handler
`ib`	`ib_object`	Immersed Boundary handler
`slices`	`slices_object`	Slice sampling for output
`rk`	`rk_object`	Runge-Kutta integrator
`weno`	`weno_object`	WENO reconstructor
`io`	`chase_io_object`	IO handler
`physics`	`chase_physics_object`	Physics — EOS, WENO basis
`ic`	`chase_ic_object`	Initial conditions
`bc`	`chase_bc_object`	Boundary conditions
`time`	`chase_time_object`	Time state and parameters
`q`	`real(R8P)(nv,…,nb)`	Conservative variables
`q_aux`	`real(R8P)(nv_aux,…,nb)`	Auxiliary variables

`initialize_common(field, filename, …)`

Parses the INI file; initialises all handlers; builds the ADAM grid, tree, and field; performs uniform refinement and pruning; binds all pointer members. Output variable names registered with ADAM IO: r_a u_a v_a w_a p_a a_a T_a E_a H_a Rfa g_a.

`adam_chase_physics_object`

Encapsulates the fluid EOS and WENO reconstruction basis selection.

Key data members

Member	Description
`nv = 5`	Conservative variable count
`nv_aux = 11`	Auxiliary variable count
`eos`	`eos_ic_object` — ideal-gas state functions (R, cv, g, …)
`weno_rec_var`	`'CONSERVATIVE'` or `'CHARACTERISTICS'`
`erw`, `elw`	Right/left eigenvector matrices for WENO projection (pointer)

WENO reconstruction basis (set via [physics] INI section, key weno_rec_var):

Value	Eigenvectors used	Description
`CONSERVATIVE`	Identity (`IERL`)	Reconstruct conservative variables directly
`CHARACTERISTICS`	`ER` / `EL`	Project to characteristics before reconstruction

Module-level procedures

Procedure	Description
`compute_auxiliary(cv, Rf, g, conservative, auxiliary)`	Conservative → auxiliary for a single cell
`compute_conservative(auxiliary, conservative)`	Auxiliary → conservative for a single cell
`compute_fluxes_conservative(sir, auxiliary, fluxes)`	Euler flux vector in direction `sir`

`adam_chase_bc_object`

Reads and stores boundary conditions for all 6 domain faces from INI sections [bc_x_min], [bc_x_max], [bc_y_min], [bc_y_max], [bc_z_min], [bc_z_max].

Constant	Value	INI keyword	Description
`BC_EXTRAPOLATION`	1	`extrapolation`	Zero-order extrapolation into ghost cells
`BC_INFLOW`	2	`inflow`	Supersonic inflow — prescribed $(ρ, u, v, w, p)$
`BC_WALL_INVISCID`	3	`wall-inviscid`	Slip wall (reflective)

Inflow state is read from INI options r, u, v, w, p (and s for a perturbation seed) under the relevant face section.

`adam_chase_ic_object`

Sets initial conditions at simulation start and after AMR grid adaptation.

IC type	INI keyword	Description
Uniform	`uniform`	Constant state with optional random velocity perturbation scaled by `q(6)`
Isentropic vortex	`isentropic-vortex`	Analytical isentropic vortex centred at `(emin_x, emin_y)`, radius from `q(6)`
Riemann problem	`riemann-problem`	Piecewise constant regions defined by bounding boxes `[emin, emax]`

INI section: [initial_conditions]. Multi-region state per box in [initial_conditions_region_N] (options r, u, v, w, p, s, emin_x/y/z, emax_x/y/z).

`adam_chase_time_object`

INI option (`[time]`)	Default	Description
`it_max`	−1	Maximum time steps (−1 = time-based termination)
`time_max`	1.0	Maximum simulation time
`CFL`	0.3	CFL stability coefficient

`adam_chase_riemann_library`

Riemann solver library providing LLF and HLL solvers and Maxwell flux functions.

Procedure	Description
`compute_riemann_maxwell_llf(sir, nv, q1, q4, f, lmax)`	LLF (Rusanov) solver; wave speed $c_{0} = 1 / \sqrt{μ_{0} ε_{0}}$
`compute_riemann_maxwell_hll(sir, nv, q1, q4, f, lmax, lmin)`	HLL solver
`compute_convective_fluxes_maxwell(sir, q, f)`	Maxwell flux vector
`compute_convective_fluxes_maxwell_div_d(sir, q, f)`	Maxwell flux with $\nabla \cdot D$ correction
`compute_convective_fluxes_maxwell_div_d_b(sir, q, f)`	Maxwell flux with $\nabla \cdot D$ and $\nabla \cdot B$ corrections

CPU backend — `chase_cpu_object`

Extends chase_common_object with working arrays and the full suite of compute methods.

Inheritance

chase_common_object          (common/)
    └── chase_cpu_object     (cpu/)   ← backend object

Additional data members

Member	Shape	Description
`dq`	`(nv, 1-ngc:ni+ngc, …, nb)`	Residual (right-hand side)
`flx`	`(nv, 1-ngc:ni+ngc, …, nb)`	Convective face fluxes in x
`fly`	`(nv, 1-ngc:ni+ngc, …, nb)`	Convective face fluxes in y
`flz`	`(nv, 1-ngc:ni+ngc, …, nb)`	Convective face fluxes in z

Methods

Lifecycle

Procedure	Purpose
`initialize(filename)`	Calls `initialize_common`; then `allocate_cpu` for working arrays
`allocate_cpu()`	Allocates `dq`, `flx`, `fly`, `flz`; zero-initialises
`simulate(filename)`	Full simulation driver: init → IC/restart → time loop → finalise

AMR

Procedure	Purpose
`amr_update()`	Iterates AMR markers; calls `adam%amr_update`; recomputes IB `phi` after each pass
`mark_by_geo(delta_fine, delta_coarse)`	Marks blocks by proximity to IB surface (`phi` sign change)
`mark_by_grad_var(grad_tol, delta_type, …)`	Marks blocks by max gradient of a field variable
`refine_uniform(levels)`	Uniformly refines all blocks
`compute_phi()`	Recomputes signed-distance IB field after grid changes
`move_phi(velocity, s)`	Translates IB solid `s` by `velocity` and updates `phi`

Immersed boundary

Procedure	Purpose
`integrate_eikonal(q)`	Runs `n_eikonal` Godunov sweeps; applies `invert_eikonal`

I/O

Procedure	Purpose
`save_xh5f([output_basename])`	Writes `q` (`r,ru,rv,rw,rE`) and `q_aux` to XH5F with Morton cell ordering
`save_restart_files()`	Saves ADAM binary restart + an XH5F snapshot
`load_restart_files(t, time)`	Reads restart binary; rebuilds MPI communication maps
`save_residuals()`	Computes per-variable L2 norm of `dq` via `MPI_ALLREDUCE(SUM)`
`save_simulation_data()`	Schedules output: XH5F snapshots, restart checkpoints, slice sampling

Slice data is written in .mat format with variable names rho rhu rhv rhw rhe.

Initial and boundary conditions

Procedure	Purpose
`set_initial_conditions()`	Delegates to `ic%set_initial_conditions`
`set_boundary_conditions(q)`	Iterates `local_map_bc_crown`; applies extrapolation or inflow BCs
`update_ghost(q [,step])`	Local inter-block copy → MPI exchange → BC application

Numerical methods

Procedure	Purpose
`compute_dt()`	CFL stability criterion (see below)
`compute_q_auxiliary(q, q_aux)`	Full-field conservative → auxiliary conversion
`compute_residuals(q, q_aux, dq)`	Ghost sync → eikonal → aux vars → fluxes → divergence
`integrate()`	Runge-Kutta time advance; same scheme dispatch as NASTO CPU
`simulate(filename)`	Full time loop driver

Time-step control

compute_dt uses a convective-only CFL bound (no viscous Fourier term):

Δ t = CFL \cdot {[max_{b, i, j, k} (\frac{| u | + a}{Δ x / 2} + \frac{| v | + a}{Δ y / 2} + \frac{| w | + a}{Δ z / 2})]}^{- 1}

A global minimum is then taken via MPI_ALLREDUCE(MPI_MIN).

Residual computation pipeline

1. update_ghost(q)                   ! sync ghost cells + apply BC
2. integrate_eikonal(q)              ! update IB distance field
3. compute_q_auxiliary(q, q_aux)     ! q → primitive/auxiliary vars
4. compute_fluxes_convective(x,y,z)  ! WENO + LLF Euler fluxes per direction
5. compute_fluxes_difference(…)      ! flux divergence → dq; IB masking

`adam_chase_common_library`

Barrel re-export module. A single use adam_chase_common_library brings all common modules into scope:

fortran

use adam_chase_bc_object
use adam_chase_common_object
use adam_chase_ic_object
use adam_chase_io_object
use adam_chase_parameters
use adam_chase_physics_object
use adam_chase_time_object

Build

bash

# GNU compiler (production)
FoBiS.py build -mode chase-gnu

# GNU compiler (debug)
FoBiS.py build -mode chase-gnu-debug

The executable adam_chase_cpu is placed in exe/.

Running

bash

# Single rank, default INI filename (input.ini)
mpirun -np 1 exe/adam_chase_cpu

# Multiple ranks, custom INI
mpirun -np 8 exe/adam_chase_cpu my_case.ini

An example INI file is provided at src/app/chase/cpu/adam_nasto.ini.

Copyrights

CHASE is part of the ADAM framework, released under the GNU Lesser General Public License v3.0 (LGPLv3).

Copyright (C) Andrea Di Mascio, Federico Negro, Giacomo Rossi, Francesco Salvadore, Stefano Zaghi.

CHASE ​

Governing equations ​

Source layout ​

Variable layout ​

Conservative variables — q(nv=5, ni, nj, nk, nb) ​

Auxiliary variables — q_aux(nv_aux=11, ni, nj, nk, nb) ​

adam_chase_common_object ​

Data members ​

initialize_common(field, filename, …) ​

adam_chase_physics_object ​

Key data members ​

Module-level procedures ​

adam_chase_bc_object ​

adam_chase_ic_object ​

adam_chase_time_object ​

adam_chase_riemann_library ​

CPU backend — chase_cpu_object ​

Inheritance ​

Additional data members ​

Methods ​

Time-step control ​

Residual computation pipeline ​

adam_chase_common_library ​

Build ​

Running ​

Copyrights ​