2016/01/22 by Tobias Ramming, Ramming, Tobias, Holger Wendland +1 · 1 citation
Engineering · #65M06 #65M12 #65M75 #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics Simulations and Interactions #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1601.06142
openalex publication_date 2016/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive a new discretisation method for first order PDEs of arbitrary\nspatial dimension, which is based upon a meshfree spatial approximation. This\nspatial approximation is similar to the SPH (smoothed particle hydrodynamics)\ntechnique and is a typical kernel-based method. It differs, however,\nsignificantly from the SPH method since it employs an Eulerian and not a\nLagrangian approach. We prove stability and convergence for the resulting\nsemi-discrete scheme under certain smoothness assumptions on the defining\nfunction of the PDE. The approximation order depends on the underlying kernel\nand the smoothness of the solution. Hence, we also review an easy way of\nconstructing smooth kernels yielding arbitrary convergence orders. Finally, we\ngive a numerical example by testing our method in the case of a one-dimensional\nBurgers equation.\n