arXiv subjects

Jiwoon Sim

Publications and source records attributed to Jiwoon Sim.

6 recordsLinked to original sources

arXiv2025

An Efficient Finite Difference-Based PML Technique for Acoustic Scattering Problems

The acoustic scattering problem is modeled by the exterior Helmholtz equation, which is challenging to solve due to both the unboundedness of the domain and the high dispersion error, known as the pollution effect. We develop high-order compact finite difference methods (FDMs) in polar coordinates to numerically solve the problem with multiple arbitrarily shaped scatterers. The unbounded domain is effectively truncated and compressed via perfectly matched layers (PMLs), while the pollution effect is handled by the high order of our method and a novel pollution minimization technique. This technique is easy to implement, rigorously proven to be effective and shows superior performance in our numerous numerical results. The FDMs we propose in regular polar coordinates achieve fourth consistency order. Yet, combined with exponential stretching and mesh refinement, we can reach sixth consistency order by slightly enlarging the stencil at certain locations. Our numerical examples demonstrate that the proposed FDMs are effective and robust under various wavenumbers, PML layer thickness and shapes of scatterers.

math.NA↗

arXiv2025

High-order, Compact, and Symmetric Finite Difference Methods for $d$-Dimensional Elliptic Equations

This paper presents compact, symmetric, and high-order finite difference methods (FDMs) for the variable Poisson equation on a $d$-dimensional hypercube. Our schemes produce symmetric linear systems: an important property that does not immediately hold for a high-order FDM. This symmetry, combined with the stencil's minimal support, keeps the storage requirements to a minimum. For the model problem considered here, the resulting linear systems are, in fact, symmetric positive definite, allowing a wide range of efficient solvers to be applied. Designing compact, symmetric, and high-order FDMs is challenging, because all overlapping stencils have to satisfy highly specific relations and central differences alone are not enough. We prove that a compact 3-point, symmetric 1D FDM on a uniform grid can achieve arbitrary consistency order. On the other hand, in the $d$-dimensional setting, where $d \ge 2$, the maximum consistency order that a compact $3^d$-point, symmetric FDM on a uniform grid can achieve is 4. If $d=2$ and the diffusion coefficient satisfies a certain derivative condition, the maximum consistency order is 6. Moreover, the compact $3^d$-point, symmetric, 4th-order FDMs for $d\ge 3$, can be conveniently expressed as a linear combination of two types of FDMs: one that depends on partial derivatives along one axis, and the other along two axes. All finite difference stencils are explicitly provided for ease of reproducibility.

math.NA↗

arXiv2025

Existence and asymptotic stability of a generic Lotka-Volterra system with nonlinear spatially heterogeneous cross-diffusion

This article considers a class of Lotka-Volterra systems with multiple nonlinear cross-diffusion, commonly known as prey-taxis models. The existence and stability of classic solutions for such systems with spatially homogeneous sources and taxis have been studied in one- or two-dimensional space, however, the proof is non-trivial for a more general setting with spatially heterogeneous predation functions and taxis coefficient functions in arbitrary dimensions. This study introduces a new weighted $L_\epsilon^p$-norm and extends some classical inequalities within this normed space. Coupled energy estimates are employed to establish initial bounds, followed by applying heat kernel properties and an advanced bootstrap process to enhance solution regularity. For stability analysis, we extend LaSalle's invariance principle to a general $ L^\infty $ setting and utilize it alongside Lyapunov functions to analyze the stability of each possible constant equilibrium. All results are achieved without introducing an extra logistic growth term for predators or imposing smallness conditions on taxis coefficients.

math.AP↗

arXiv2025

Shifts of dominant personality and spatial pattern formation due to spatially heterogeneous pollution

Personality traits, such as boldness and shyness, play a significant role in shaping the survival strategies of animals. Industrial pollution has long posed serious threats to ecosystems and is typically distributed heterogeneously. However, how animals with different personalities respond to spatially heterogeneous pollution remains largely unexplored. In this study, we introduce a prey-taxis model with nonlinear cross-diffusion to examine population dynamics in such environments. The global existence of classical solutions is established by deriving initial bounds through energy estimates and improving solution regularity via heat kernel properties and a bootstrap process. Our findings reveal that behavior, population structure, and spatial distribution are heavily influenced by pollution. Bold individuals maintain a competitive advantage in pollution-free or very low-toxin environments, whereas shy individuals become dominant in regions with low to moderate toxin levels. In highly polluted areas, no populations can survive. The spatial pattern of the population is also closely tied to the distribution of toxins. Grazers tend to move along toxin gradient and exhibit periodic behavior. As toxin concentrations rise, aggregation behavior becomes increasingly pronounced across all species. Interestingly, the total population in polluted areas may initially increase when toxin levels are low to moderate, but eventually declines, leading to extinction as toxin levels continue to rise.

q-bio.PE↗

arXiv2025

Convergent Sixth-order Compact Finite Difference Method for Variable-Coefficient Elliptic PDEs in Curved Domains

Finite difference methods (FDMs) are widely used for solving partial differential equations (PDEs) due to their relatively simple implementation. However, they face significant challenges when applied to non-rectangular domains and in establishing theoretical convergence, particularly for high-order schemes. In this paper, we focus on solving the elliptic equation $-\nabla \cdot (a \nabla u) = f$ in a two-dimensional curved domain $\Omega$, where the diffusion coefficient $a$ is variable and smooth. We propose a sixth-order $9$-point compact FDM on uniform Cartesian grids within the domain, not relying on ghost points or information outside $\overline{\Omega}$. All the boundary stencils near $\partial \Omega$ have at most $6$ different configurations and use at most $8$ grid points inside $\Omega$. We rigorously establish the sixth-order convergence of the numerically approximated solution $u_h$ in the $\infty$-norm. Additionally, we derive a gradient approximation $\nabla u$ directly from $u_h$ without solving auxiliary equations. This gradient approximation achieves proven accuracy of order $5 + \frac{1}{q}$ in the $q$-norm for all $1 \leq q \leq \infty$ (with a logarithmic factor $\log h$ for $1 \leq q < 2$). To validate our proposed sixth-order compact finite different method, we provide several numerical examples that illustrate the sixth-order accuracy and computational efficiency of both the numerical solution and the gradient approximation for solving elliptic PDEs in curved domains.

math.NA↗

arXiv2024

Uniqueness of the inverse source problem for fractional diffusion-wave equations

This study addresses the inverse source problem for the fractional diffusion-wave equation, characterized by a source comprising spatial and temporal components. The investigation is primarily concerned with practical scenarios where data is collected subsequent to an incident. We establish the uniqueness of either the spatial or the temporal component of the source, provided that the temporal component exhibits an asymptotic expansion at infinity. Taking anomalous diffusion as a typical example, we gather the asymptotic behavior of one of the following quantities: the concentration on partial interior region or at a point inside the region, or the flux on partial boundary or at a point on the boundary. The proof is based on the asymptotic expansion of the solution to the fractional diffusion-wave equation. Notably, our approach does not rely on the conventional vanishing conditions for the source components. We also observe that the extent of uniqueness is dependent on the fractional order.

math.NA↗