From equations to a numerical solver

Turn continuous equations into algebra

Approach	Basic idea	What to ask
Finite volume	Integrate conservation laws over control volumes and approximate their face fluxes.	How are inviscid and viscous fluxes reconstructed? What limiter or dissipation is used?
Finite difference	Approximate derivatives from neighboring sample values using a stencil.	How does the stencil handle boundaries, curvilinear mappings and shocks?
Finite element	Express the solution using element basis functions and enforce a weak form.	Which spaces and stabilization are used?
Discontinuous Galerkin and flux reconstruction	Use element-local polynomial representations and couple through interface fluxes or corrections.	What order, shock treatment and curved-element geometry are supported?

Time integration is a separate choice. The advective Courant number is roughly $U\Delta t/\Delta x$ in one dimension. Explicit compressible schemes also account for acoustic wave speed. Implicit stepping can ease stability limits while physical time accuracy still depends on time-step size.

Lattice Boltzmann is a different equation family

LBM advances discrete particle distribution functions through collision and streaming. Fluid variables are obtained from moments of those distributions. In an appropriate hydrodynamic limit the method recovers continuum flow behavior. For the simple single-relaxation-time lattice BGK model, the collide-and-stream step without forcing is

$$f_i(\mathbf x+\mathbf c_i\Delta t,t+\Delta t) =f_i(\mathbf x,t)-\omega_{\mathrm{LB}}[f_i(\mathbf x,t)-f_i^{\mathrm{eq}}(\mathbf x,t)].$$

Here $f_i$ is the population for discrete velocity $\mathbf c_i$ and $\omega_{\mathrm{LB}}$ is the relaxation parameter. Density and momentum come from $\sum_i f_i$ and $\sum_i f_i\mathbf c_i$. Commercial products can use different collision operators. Read the documentation for allowable Mach numbers, wall treatment and turbulence modeling. An LBM product can still use a subgrid model such as LES.

Reduced-order aircraft tools

Vortex-lattice and potential-flow panel methods are useful for early loading estimates within their assumptions. Airfoil tools such as XFOIL couple an inviscid outer solution to an integral boundary-layer model. They can be useful alongside volume CFD for early design comparisons.

Sources

• NASA · CFD Vision 2030, discretizations and solvers
• PyFR · high-order flux reconstruction
• MIT · XFOIL
• Qian, d’Humières and Lallemand · lattice BGK model
• OpenLB · lattice Boltzmann library