arXiv subjects

Kiwoong Kwon

Publications and source records attributed to Kiwoong Kwon.

6 recordsLinked to original sources

arXiv2026

A posteriori error estimates for a finite volume approximation of a local sensing Keller-Segel model

A local sensing Keller-Segel model, in which cells adjust their motility to the local chemoattractant rather than its gradient, and a semi-implicit finite volume scheme for its numerical approximation are studied. Computable a posteriori bounds for the approximation error measured in terms of the $L^\infty(0, T; H^{-1}(Ω))$ and $L^2(0, T; L^2(Ω))$ norms of the cell density are derived based on a new stability estimate and suitable reconstructions of the flux and the flux potential of the numerical solution. Optimal scaling of the estimates is numerically demonstrated in 1D and 2D experiments. Moreover, an adaptive mesh refinement method steered by the estimator is proposed for the 1D case, and its efficiency is numerically verified.

math.NA↗

arXiv2026

A structure-preserving finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility

We construct and analyze a fully discrete finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility. The scheme combines a standard two-point flux approximation of diffusion, the L1 approximation of the Caputo derivative on a uniform time grid, and a semiconvex splitting of the potential, and it discretizes the inverse mobility by a secant average multiplying the entire L1 derivative. The key estimate is a lower bound for the accumulated pairing of this weighted L1 derivative with the solution increments in terms of the accumulated L1 quadratic form. Using this lower bound, we derive a discrete energy estimate and the time regularity required for compactness. For twice continuously differentiable semiconvex potentials satisfying sign conditions outside $(-1,1)$ and initial data in $H^1(Ω)$ taking values in $[-1,1]$, we prove existence of at least one fully discrete solution on every admissible finite volume mesh and every uniform time grid. Every such solution satisfies the maximum bound principle and a discrete energy bound relative to the initial energy, without a time-step restriction. In up to three space dimensions, under a mesh regularity condition with a constant independent of the mesh size, we prove that any sequence of fully discrete solutions has a subsequence converging strongly in $L^2(Ω\times(0,T))$ to a weak solution of the continuous problem as the mesh size and time step tend to zero. As a consequence, we obtain existence of a weak solution in one and two space dimensions, and in three dimensions whenever such meshes exist.

math.NA↗

arXiv2026

A Posteriori Error Analysis of Runge-Kutta Discontinuous Galerkin Schemes with SIAC Post-Processing for Nonlinear Convection-Diffusion Systems

We develop reliable a posteriori error estimators for fully discrete Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems endowed with a convex entropy in multiple spatial dimensions on the flat torus T^d, with a focus on the convection-dominated regime. In order to use the relative entropy method, we reconstruct the numerical solution via tensor-product Smoothness-Increasing Accuracy-Conserving (SIAC) filtering which has superconvergence properties. We then derive reliable a posteriori error estimators for the difference between the entropy weak solution and the reconstruction, with constants that are uniform in the vanishing viscosity limit. Our numerical experiments show that the a posteriori error bounds converge with the same order as the error of the reconstructed numerical solution.

math.NA↗

arXiv2025

A posteriori analysis for nonlinear convection-diffusion systems

This work provides reliable a posteriori error estimates for Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems. The classes of systems we study are quite general with a focus on convection-dominated and degenerate parabolic problems. Our a posteriori error bounds are valid for a family of discontinuous Galerkin spatial discretizations and various temporal discretizations that include explicit and implicit-explicit time-stepping schemes, popular tools for practical simulations of this class of problem. We prove that our estimators provide reliable upper bounds for the error of the numerical method and present numerical evidence showing that they achieve the same order of convergence as the error. Since one of our main interests is the convection dominant case, we also track the dependence of the estimator on the viscosity coefficient.

math.NA↗

arXiv2024

A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model

We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results.

math.NA↗

arXiv2024

Relative entropy technique in terms of position and momentum and its application to Euler-Poisson system

This paper presents a systematic study of the relative entropy technique for compressible motions of continuum bodies described as Hamiltonian flows. While the description for the classical mechanics of $N$ particles involves a Hamiltonian in terms of position and momentum vectors, that for the continuum fluid involves a Hamiltonian in terms of density and momentum. For space dimension $d\ge 2$, the Hamiltonian functional has a non-convex dependency on the deformation gradient or placement map due to material frame indifference. Because of this, the applicability of the relative entropy technique with respect to the deformation gradient or the placement map is inherently limited. Despite these limitations, we delineate the feasible applications and limitations of the technique by pushing it to its available extent. Specifically, we derive the relative Hamiltonian identity, where the Hamiltonian takes the position and momentum field as its primary and conjugate state variables, all within the context of the referential coordinate system that describes the motion. This approach, when applicable, turns out to yield rather strong stability statements. As instances, we consider Euler-Poisson systems in one space dimension. For a specific pressureless model, we verify non-increasing $L^2$ state differences before the formation of $δ$-shock. In addition, weak-strong uniqueness, stability of rarefaction waves, and convergence to the gradient flow in the singular limit of large friction are shown. Depending on the presence or absence of pressure, assumptions are made to suitably accommodate phenomena such as $δ$-shocks, vacuums, and shock discontinuities in the weak solutions.

math.AP↗