arXiv subjects

Bojan Popov

Publications and source records attributed to Bojan Popov.

11 recordsLinked to original sources

arXiv2024

First-order greedy invariant-domain preserving approximation for hyperbolic problems: scalar conservation laws, and p-system

The paper focuses on first-order invariant-domain preserving approximations of hyperbolic systems. We propose a new way to estimate the artificial viscosity that has to be added to make explicit, conservative, consistent numerical methods invariant-domain preserving and entropy inequality compliant. Instead of computing an upper bound on the maximum wave speed in Riemann problems, we estimate a minimum wave speed in the said Riemann problems such that the approximation satisfies predefined invariant-domain properties and predefined entropy inequalities. This technique eliminates non-essential fast waves from the construction of the artificial viscosity, while preserving pre-assigned invariant-domain properties and entropy inequalities.

math.NA↗

arXiv2022

Robust second-order approximation of the compressible Euler equations with an arbitrary equation of state

This paper is concerned with the approximation of the compressible Euler equations supplemented with an arbitrary or tabulated equation of state. The proposed approximation technique is robust, formally second-order accurate in space, invariant-domain preserving, and works for every equation of state, tabulated or analytic, provided the pressure is nonnegative. An entropy surrogate functional that grows across shocks is proposed. The numerical method is verified with novel analytical solutions and then validated with several computational benchmarks seen in the literature.

math.NA↗

arXiv2021

On the implementation of a robust and efficient finite element-based parallel solver for the compressible Navier-Stokes equations

This paper describes in detail the implementation of a finite element technique for solving the compressible Navier-Stokes equations that is provably robust and demonstrates excellent performance on modern computer hardware. The method is second-order accurate in time and space. Robustness here means that the method is proved to be invariant domain preserving under the hyperbolic CFL time step restriction, and the method delivers results that are reproducible. The proposed technique is shown to be accurate on challenging 2D and 3D realistic benchmarks.

math.NA↗

arXiv2021

Hyperbolic relaxation technique for solving the dispersive Serre-Green-Naghdi Equations with topography

The objective of this paper is to propose a hyperbolic relaxation technique for the dispersive Serre-Green-Naghdi equations (also known as the fully non-linear Boussinesq equations) with full topography effects introduced in Green, A.E. and Naghdi, P.M. (J. Fluid Mech., 78, 237-246, 1976) and Seabra-Santos el al (J. Fluid Mec.h, 176, 117-134, 1997). This is done by revisiting a similar relaxation technique introduced in Guermond el al (J. Comput. Phys., 399, 108917, 2019) with partial topography effects. We also derive a family of analytical solutions for the one-dimensional dispersive Serre-Green-Naghdi equations that are used to verify the correctness the proposed relaxed model. The method is then numerically illustrated and validated by comparison with experimental results.

math.NA↗

arXiv2020

Second-order invariant domain preserving approximation of the compressible Navier--Stokes equations

We present a fully discrete approximation technique for the compressible Navier-Stokes equations that is second-order accurate in time and space, semi-implicit, and guaranteed to be invariant domain preserving. The restriction on the time step is the standard hyperbolic CFL condition, ie $τ\lesssim \mathcal{O}(h)/V$ where $V$ is some reference velocity scale and $h$ the typical meshsize.

math.NA↗

arXiv2019

Positive asymptotic preserving approximation of the radiation transport equation

We introduce a (linear) positive and asymptotic preserving method or solving the one-group radiation transport equation. The approximation in space is discretization agnostic: the space approximation can be done with continuous or discontinuous finite elements (or finite volumes, or finite differences). The method is first-order accurate in space. This type of accuracy is coherent with Godunov's theorem since the method is linear. The two key theoretical results of the paper are Theorem~4.4 and Theorem~4.8. The method is illustrated with continuous finite elements. It is observed to converge with the rate $\calO(h)$ in the $L^2$-norm on manufactured solutions, and it is $\calO(h^2)$ in the diffusion regime. Unlike other standard techniques, the proposed method does not suffer from overshoots at the interfaces of optically thin and optically thick regions.

math.NA↗

arXiv2018

Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems

We introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving. The method is discretization-independent provided elementary symmetry and skew-symmetry properties are satisfied by the scheme. The method is formally first-order accurate in space. A series of higher-order methods is also introduced. When these methods violate the invariant domain properties, they are corrected by a limiting technique that we call convex limiting. After limiting, the resulting methods satisfy all the invariant domain properties that are imposed by the user (see Theorem~7.24). A key novelty is that the bounds that are enforced on the solution at each time step are necessarily satisfied by the low-order approximation.

math.NA↗

arXiv2017

Second-order invariant domain preserving approximation of the Euler equations using convex limiting

A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.

math.NA↗

arXiv2015

Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations

This paper is concerned with the construction of a fast algorithm for computing the maximum speed of propagation in the Riemann solution for the Euler system of gas dynamics with the co-volume equation of state. The novelty in the algorithm is that it stops when a guaranteed upper bound for the maximum speed is reached with a prescribed accuracy. The convergence rate of the algorithm is cubic and the bound is guaranteed for gasses with the co-volume equation of state and the heat capacity ratio $\gamma$ in the range $(1,5/3]$

math.NA↗

arXiv2015

Invariant domains and first-order continuous finite element approximation for hyperbolic systems

We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant sets containing the initial data is an invariant domain for the method. The invariant domain property is proved for any hyperbolic system provided a CFL condition holds. The solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system. The method is formally first-order accurate in space and can be made high-order in time by using Strong Stability Preserving algorithms. This technique extends to continuous finite elements the work of \cite{Hoff_1979,Hoff_1985}, and \cite{Frid_2001}.

math.NA↗

arXiv2012

Viscous regularization of the Euler equations and entropy principles

This paper investigates a general class of viscous regularizations of the compressible Euler equations. A unique regularization is identified that is compatible with all the generalized entropies a la Harten and satisfies the minimum entropy principle. A connection with a recently proposed phenomenological model by H. Brenner is made.

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