arXiv subjects

Peter Frolkovic

Publications and source records attributed to Peter Frolkovic.

3 recordsLinked to original sources

arXiv2026

Compact implicit high-resolution numerical scheme for multicomponent transport problems with Langmuir sorption

We present a compact implicit high-resolution finite volume scheme for one-dimensional multicomponent transport problems with nonlinear Langmuir sorption. The method combines a second order compact implicit discretization with WENO reconstruction and a time limiter for the treatment of steep gradients and discontinuities. A special attention is given to the Courant number used in the limiter, where local characteristic Courant numbers based on the eigenvalues of the inverse retardation matrix are considered. The compact implicit schemes are compared with analogous explicit first and second order methods. Numerical experiments show second order convergence for smooth solutions and demonstrate that the implicit approach can be used with significantly larger Courant numbers. The method is also applied to a three-component displacement chromatography problem with sharp concentration fronts.

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arXiv2025

Compact implicit high resolution numerical method for solving transport problems with sorption isotherms

This study investigates numerical methods to solve nonlinear transport problems characterized by various sorption isotherms with a focus on the Freundlich type of isotherms. We describe and compare second order accurate numerical schemes, focusing on implicit methods, to effectively model transport phenomena without stability restriction on the choice of time steps. Furthermore, a high resolution form of the method is proposed that limits a priori the second order accurate scheme towards first order accuracy to keep the values of numerical solutions in a physically acceptable range. Through numerical experiments, we demonstrate the effectiveness of high resolution methods in minimizing oscillations near discontinuities, thereby enhancing solution plausibility. The observed convergence rates confirm that the second order accurate schemes achieve expected accuracy for smooth solutions and that they yield significant improvements when compared with the results of the first order scheme. As the computational cost of the compact implicit method seems to be comparable to similar explicit ones with a clear profit of unconditional stability, this research provides a practical tool toward numerical simulations of nonlinear transport phenomena applicable in various fields such as contaminant transport in porous media or column liquid chromatography.

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arXiv2024

Numerical solution of two dimensional scalar conservation laws using compact implicit numerical schemes on Cartesian meshes

This paper deals with the numerical solution of conservation laws in the two dimensional case using a novel compact implicit time discretization that enables applications of fast algebraic solvers. We present details for the second order accurate parametric scheme based on the finite volume method including simple variants of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) approximations. To avoid oscillatory numerical solutions for large time steps, we propose limiting in time which is provable non-oscillatory in the case of linear advection equation with variable velocity. We present numerical experiments for representative linear and nonlinear problems.

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