Lagrangian approach

Lagrangian approach naturally observes movement, and thus advection of flow. Points that describe continuum move along streamlines and transport their properties.

Eulerian approach statically observes some locations while the medium flows through them, which requires the addition of non-linear operator to transport equations (u · ∇).

RIP Courant–Friedrichs–Lewy

The classical CFL condition does not apply to Lagrangian methods, which may have large time-steps with stable advection.

Meshless FD framework

Due to freedom of movement, the Lagrangian method is ideal to be described by meshless (or mesh-free) framework.

Novel 2nd order Finite-Difference (FD) spatial operators are introduced using Weighted Least-Squares (WLS) to accurately solve Partial Differential Equation (PDEs).

Incompressible flows

Navier-Stokes equations (NSE) are solved using velocity-pressure decoupling. The solver can iterate multiple times through the pressure and momentum equations per time-step to keep the simulation accurate and stable (similar to PIMPLE algorithm in the FVM).

The volume conservation is unconditionally stable and handles potentially compressible situations (as shown on an exaggerated example).

Fluid-structure interaction

Users do not have to deal with volumetric meshing, but they also do not have to deal with boundary surfaces meshing or conversion to meshless points.

The solver implements novel treatment of boundary conditions, which are directly applied on imported triangulated geometry.

This makes transfer of loads and coupling with some FEM solver straightforward, in order to simulate FSI with large deformations and movements.

High-performance computing

The solver is completely parallelized, i.e. runs on multi-core and many-core devices. The only bottleneck is when the solver needs to transfer data (due to outputting results to disk, or waiting for some coupled solver).

As GPUs have many parallel execution units, they are ideal candidate for solving repetitive tasks in CFD. Rhoxyz simulations may run more than 20x faster on the modern GPU, than on the modern CPU.

Graphical user interface

The solver comes with a desktop application for setting up a simulation, controlling the simulation running process, and real-time visualisation of the results while the simulation is running.

The simulation results may be exported and loaded for more complex post-processing, e.g. using ParaView.


Basic, J. Development of numerical model for green water loading by coupling the mesh based flow models with the meshless models. PhD dissertation, University of Zagreb (2019).

Basic, J., Blagojevic, B., Andrun, M. & Degiuli, N. A Lagrangian Finite Difference Method for Sloshing: Simulations and Comparison with Experiments. in Proceedings of the Twenty-ninth International Ocean and Polar Engineering Conference 3412–3418 (2019).

Basic, J., Blagojevic, B., Ban, D., Ljubenkov, B. & Degiuli, N. Lagrangian Finite Difference Method for Violent Fluid–Structure Interaction. in Proceedings of the 32nd Symposium on Naval Hydrodynamics 14 (2018).

Basic, J., Degiuli, N. & Ban, D. A class of renormalised meshless Laplacians for boundary value problems. J. Comput. Phys. 354, 269–287 (2018).

Bašić, J., Degiuli, N. & Ban, D. Renormalised Lagrangian method for water entry impact simulation. in Proceedings of the 11th Symposium on High Speed Marine Vehicles 8 (2017).