MVP_CFD — Advection-Diffusion Solver

A numerical solver in C++ for the advection-diffusion equation, currently supporting 1D and 2D formulations. Designed as a modular, extensible framework with future support for RANS equations planned.

Features

Spatial Schemes

Scheme	Order	Notes
Central Difference	2nd	May produce spurious oscillations at high Pe
Upwind	1st	Stable and monotone, recommended for advection-dominated flows
QUICK	3rd	Higher accuracy, requires finer grids

Time Schemes

Scheme	Order	Notes
Explicit Euler	1st	Simple, subject to CFL and diffusion stability limits
Runge-Kutta 4	4th	High accuracy, recommended for precision simulations

Boundary Conditions

• Dirichlet — fixed value at boundaries
• Neumann — fixed derivative at boundaries
• Periodic — u(0) = u(L)

Initial Conditions

• Gaussian — smooth bell-shaped profile (1D only)
• Square wave — sharp step profile (1D & 2D)
• Sinusoidal — smooth periodic oscillations (1D only)

Dimensionless Numbers

Courant-Friedrichs-Lewy (CFL)

CFL = c·Δt/Δx ≤ 1   (explicit schemes)

Diffusion Number

Diff = D·Δt/Δx² ≤ 0.5   (explicit schemes)

Peclet Number

Pe = c·L/D

• Pe << 1 — diffusion dominated
• Pe >> 1 — advection dominated
• Pe ~ 1 — mixed regime

Build

The project uses a Makefile with three build configurations:

make                  # Release build (optimized, -O3)
make BUILD=debug      # Debug build (-g, no optimization)
make BUILD=asan       # AddressSanitizer build (memory/UB checks)

Executables produced:

• solver — release
• solver_debug — debug
• solver_asan — asan

make run              # Build (release) and run with config.cfg
make clean            # Remove objects and executables
make cleanall         # Also remove output files
make help             # Show all available targets

Usage

The quickest way to run is:

make run

This builds the project (if needed) and runs it automatically with config.cfg. To use a different config file, run the executable directly:

./solver <config_file>
# example:
./solver AdvDiff1D.cfg

Configuration File

The solver is fully configured via a .cfg text file. Lines starting with # are comments.

# MVP CFD solver configuration file

# Solver type
SOLVER = BURGERS_2D 	 			# ADVECTION_DIFFUSION_1D | ADVECTION_DIFFUSION_2D | BURGERS_2D

# Grid options
GRID_POINTS_X = 201
GRID_POINTS_Y = 201
DOMAIN_LENGHT_X = 2.0
DOMAIN_LENGHT_Y = 2.0

# Simulation settings
DIFFUSION_COEFF = 0.0
TIME_STEP = 0.001
INITIAL_TIME = 0.0
END_TIME = 3.0

# Spatial and Time schemes
SPATIAL_SCHEME = QUICK   			# UPWIND | CENTRAL | QUICK
TIME_SCHEME = RK4					# EULER_EXPLICIT | RK4

# Initial Conditions Settings
INITIAL_CONDITIONS = SQUARE_WAVE

# Possible value for INITIAL_CONDITIONS:
#   GAUSSIAN    (1D only)  → requires: AMPLITUDE_IC, X0_IC, SIGMA_IC
#   SQUARE_WAVE (1D & 2D)  → requires: AMPLITUDE_IC, BASELINE_VALUE_IC, RANGE_START_IC, RANGE_END_IC (1D)
#                                       AMPLITUDE_IC, BASELINE_VALUE_IC, RANGE_START_X_IC, RANGE_END_X_IC,
#                                                     RANGE_START_Y_IC, RANGE_END_Y_IC (2D)
#   SINUSOIDAL  (1D only)  → requires: AMPLITUDE_IC
#   CONSTANT    (1D & 2D)  → requires: AMPLITUDE_IC

============================ Boundary Condition settings ==============================

# Advection-Diffusion solver (scalar field: e.g. temperature, concentration)
# Supports 1D and 2D configurations depending on grid setup
#
# Boundary condition is global (same type applied to all boundaries)
# Allowed values: DIRICHLET | NEUMANN | PERIODIC
#
# BC_* values define scalar boundary values (used only for DIRICHLET and NEUMANN)

BOUNDARY_CONDITIONS = PERIODIC  # DIRICHLET | NEUMANN | PERIODIC

BC_LEFT = 0.0
BC_RIGHT = 0.0
BC_TOP = 0.0
BC_BOTTOM = 0.0

=======================================================================================

# Burgers equation solver (2D vector field: U, V)
# Boundary conditions are defined per face (bottom, top, left, right)
#
# Supports Dirichlet (D) and Neumann (N) boundary conditions
#
# For periodic boundary conditions, set:
#   BOUNDARY_CONDITION = PERIODIC
# In this case, periodicity overrides individual boundary type settings.
#
# Example: lid-driven configuration (top lid moving in x-direction)

BOUNDARY_CONDITIONS = DIRICHLET  # DIRICHLET | NEUMANN | PERIODIC

BOUNDARY_CONDITIONS_VALUES_U = {0.0, 1.0, 0.0, 0.0}
BOUNDARY_CONDITIONS_VALUES_V = {0.0, 0.0, 0.0, 0.0}

=======================================================================================

# Boundary Conditions Settings for Incompressible Navier Stokes solver
# Layout: { bottom, top, left, right }
#
# BOUNDARY_CONDITIONS_VALUES_X defines the prescribed values for each boundary face.
# Example: BOUNDARY_CONDITIONS_VALUES_U = {0.0, 1.0, 0.0, 0.0}
#
# BOUNDARY_CONDITIONS_TYPES_X defines the type of boundary condition per face.
# Allowed values: Dirichlet (D), Neumann (N)
# Example: BOUNDARY_CONDITIONS_TYPES_U = {D, N, N, N}
#
# At least one Dirichlet boundary condition is required to ensure a well-posed problem
#
# For periodic boundary conditions, set:
#   BOUNDARY_CONDITION = PERIODIC
# In this case, periodicity overrides individual boundary type settings.
#
# Otherwise set:
#   BOUNDARY_CONDITION = NON_PERIODIC
# and use BOUNDARY_CONDITIONS_TYPES_X to define D/N conditions.
#
# Example: lid-driven cavity
# Top wall moves right (u = 1.0), all other walls are no-slip (u = 0.0)

BOUNDARY_CONDITIONS_VALUES_U = {0.0, 1.0, 0.0, 0.0}
BOUNDARY_CONDITIONS_VALUES_V = {0.0, 0.0, 0.0, 0.0}
BOUNDARY_CONDITIONS_VALUES_P = {0.0, 0.0, 0.0, 0.0}

BOUNDARY_CONDITIONS_TYPES_U = {D,D,D,D}
BOUNDARY_CONDITIONS_TYPES_V = {D,D,D,D}
BOUNDARY_CONDITIONS_TYPES_P = {N,D,N,N}

BOUNDARY_CONDITION = NON_PERIODIC

=======================================================================================

#Output flag
VERBOSE = true

#Results
OUTPUT_DIR = results/
OUTPUT_FILE = output
OUTPUT_FREQUENCY = 20

Results and Visualization

1D case: Results are saved as .csv files in the results/ directory:

• mesh.csv — grid coordinates
• output.csv — solution u at each saved time step (one row per step)

Visualization is handled by a Python script inside the results/ directory: visualize.py

To visualize the results simply run:

pyhton3 visualize.py

2D case: Results are saved as .vtu files in the results/ directory. The .pvd file is used to visualize all the results saved in the .vtu files.

To visualize the results open the .pvd file using Paraview.

Roadmap

• 1D Advection-Diffusion solver
• 2D Advection-Diffusion solver
• 2D Burgers solver
• RANS equations
• Implicit schemes
• Non-uniform grids
• OpenMP/MPI parallelisation
• VTK output for Paraview

References

1. LeVeque, R. J. (2002). Finite Volume Methods for Hyperbolic Problems
2. Lorena A. Barba CFD Python: 12 steps to Navier-Stokes
3. Politecnico di Milano. Lecture notes from university courses.
4. Tom-Robin Teschner. cfd.university

License

Free for educational and research use.