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Numerical Methods for Scientific Computing

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: closed (30 June 2026) | Viewed by 6995

Editors

Prof. Dr. Ching-Sung Liu

E-Mail Website
Guest Editor
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
Interests: tensor/matrix analysis and computations; optimization; theory and algorithms; numerical analysis and scientific computing
Prof. Dr. Xiaobing Feng

E-Mail Website
Guest Editor
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA
Interests: computational and applied mathematics; numerical analysis; scientific computing; numerical solutions of partial differential equations; uncertainty quantification; fractional calculus; fractional/nonlocal differential equations; deep neural networks; high-dimensional computation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The Special Issue, “Numerical Methods for Scientific Computing”, focuses on the application and development of computational techniques for solving scientific problems and delves into various numerical methods employed in fields such as physics, engineering, mathematics, and more. The Special issue explores topics including numerical differential equations, numerical optimization, numerical linear algebra, numerical multilinear algebra, eigenvalue problems and nonlinear eigenvalue problems, and matrix and tensor approximation. The emphasis is placed on the theoretical foundations, algorithmic advancements, and practical applications of these numerical techniques. By addressing these areas, this Special Issue aims to contribute to the advancement of scientific computing and its interdisciplinary applications.

Prof. Dr. Ching-Sung Liu
Prof. Dr. Xiaobing Feng
Guest Editors

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Keywords

  • numerical methods
  • numerical differential equations
  • numerical optimization
  • numerical linear algebra
  • numerical multilinear algebra
  • eigenvalue problems
  • nonlinear eigenvalue problems
  • matrix approximation
  • tensor approximation
  • bisection method
  • Newton’s method
  • secant method
  • integration
  • midpoint rule
  • trapezoidal rule
  • adaptive integration
  • initial value problem
  • euler method
  • second-order runge-kutta methods
  • system of differential equations
  • boundary value problems
  • linear algebra
  • gaussian elimination
  • finite difference approximation
  • iterative methods
  • piecewise linear interpolation
  • multidimensional interpolation
  • least-squares approximation

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Published Papers (5 papers)

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Research

27 pages, 2341 KB  
Article
An Improved Conservative Hybrid Method with Adaptive Mesh Refinement for Passive-Scalar Transport on Deforming Interfaces
by Yu Fan and Chunyan Liu
Mathematics 2026, 14(7), 1146; https://doi.org/10.3390/math14071146 - 29 Mar 2026
Viewed by 523
Abstract
This paper presents an improved hybrid Eulerian–Lagrangian framework, which has been augmented with an adaptive mesh refinement technique, for simulating passive scalar transport on deforming interfaces. We capture interface deformation using an Eulerian level-set method while solving the interfacial transport equation with a [...] Read more.
This paper presents an improved hybrid Eulerian–Lagrangian framework, which has been augmented with an adaptive mesh refinement technique, for simulating passive scalar transport on deforming interfaces. We capture interface deformation using an Eulerian level-set method while solving the interfacial transport equation with a single-layer smoothed particle hydrodynamics method. As a result, the proposed hybrid approach combines the high efficiency of the Eulerian formulation with the strict mass conservation property of smoothed particle hydrodynamics method. To further accelerate the simulations, we employ adaptive mesh refinement for the Eulerian solver and restrict particles to the finest refinement level. To mitigate Lagrangian particle clustering, we adopt a remeshing procedure that generates particle distributions adapted to the local interface geometry on the finest mesh. This remeshing also enables accurate, mass-conservative reconstruction of the interfacial concentration field. Moreover, by incorporating an adaptive remeshing strategy, we tune the remeshing frequency to balance computational cost and accuracy. The accuracy and robustness of the proposed method are demonstrated through a suite of benchmark test cases. Additionally, we evaluate the effectiveness of adaptive mesh refinement through benchmark test cases, verifying its compatibility with the interfacial smoothed particle hydrodynamics method and quantifying the resulting speedup. Full article
(This article belongs to the Special Issue Numerical Methods for Scientific Computing)
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17 pages, 5402 KB  
Article
Fourth-Order Compact Finite-Difference Scheme with Discrete Sine Transform for Solving 2D Heat Conduction Equation with DBCs
by Chunming Liu and Xiaozhong Tong
Mathematics 2026, 14(6), 949; https://doi.org/10.3390/math14060949 - 11 Mar 2026
Viewed by 619
Abstract
Finite-difference approaches are widely employed to solve partial differential equations in numerous practical applications. However, their computational efficiency is often limited by the need to solve linear systems through matrix inversion or iterative solvers, a challenge that is particularly acute in high-dimensional problems. [...] Read more.
Finite-difference approaches are widely employed to solve partial differential equations in numerous practical applications. However, their computational efficiency is often limited by the need to solve linear systems through matrix inversion or iterative solvers, a challenge that is particularly acute in high-dimensional problems. Consequently, there is a growing demand for methods that ensure both high accuracy and computational efficiency. To address the two-dimensional (2D) heat conduction problem, we propose a novel hybrid technique that integrates a fourth-order implicit compact finite-difference approach with the discrete sine transform (DST). The incorporation of the DST significantly reduces the computational burden associated with solving the heat conduction equation on large grids. Detailed numerical experiments were conducted to evaluate this solver for 2D heat conduction equations subject to homogeneous Dirichlet boundary conditions (DBCs). The results demonstrate that the proposed method not only achieves substantial reductions in computational cost but also maintains a high level of numerical accuracy. All numerical experiments were performed on a computer running MATLAB R2024b. Full article
(This article belongs to the Special Issue Numerical Methods for Scientific Computing)
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17 pages, 405 KB  
Article
Shared-Pole Carathéodory–Fejér Approximations for Linear Combinations of φ-Functions
by Awad H. Al-Mohy
Mathematics 2025, 13(24), 3985; https://doi.org/10.3390/math13243985 - 14 Dec 2025
Viewed by 804
Abstract
We develop a shared denominator Carathéodory–Fejér (CF) method for efficiently evaluating linear combinations of φ-functions for matrices whose spectrum lies in the negative real axis, as required in exponential integrators for large stiff ODE systems. This entire family is approximated with a [...] Read more.
We develop a shared denominator Carathéodory–Fejér (CF) method for efficiently evaluating linear combinations of φ-functions for matrices whose spectrum lies in the negative real axis, as required in exponential integrators for large stiff ODE systems. This entire family is approximated with a single set of poles (a common denominator). The shared pole set is obtained by assembling a stacked Hankel matrix from Chebyshev boundary data for all target functions and computing a single SVD; the zeros of the associated singular-vector polynomial, mapped via the standard CF slit transform, yield the poles. With the poles fixed, per-function residues and constants are recovered by a robust least squares fit on a suitable grid of the negative real axis. For any linear combination of resolvent operators applied to right-hand sides, the evaluation reduces to one shifted linear solve per pole with a single combined right-hand side, so the dominant cost matches that of computing a single φ-function action. Numerical experiments indicate geometric convergence at a rate consistent withHalphen’s constant, and for highly stiff problems our algorithm outperforms existing Taylor and Krylov polynomial-based algorithms. Full article
(This article belongs to the Special Issue Numerical Methods for Scientific Computing)
17 pages, 2157 KB  
Article
A Hybrid DST-Accelerated Finite-Difference Solver for 2D and 3D Poisson Equations with Dirichlet Boundary Conditions
by Jing Pei and Xiaozhong Tong
Mathematics 2025, 13(17), 2776; https://doi.org/10.3390/math13172776 - 28 Aug 2025
Cited by 4 | Viewed by 1996
Abstract
Finite-difference methods are widely used to solve partial differential equations in diverse practical applications. Despite their prevalence, the computational efficiency of these methods encounters limitations due to the need to solve linear equation systems through matrix inversion or iterative solver, which is particularly [...] Read more.
Finite-difference methods are widely used to solve partial differential equations in diverse practical applications. Despite their prevalence, the computational efficiency of these methods encounters limitations due to the need to solve linear equation systems through matrix inversion or iterative solver, which is particularly challenging in scenarios involving high dimensions. The demand for numerical methods with high accuracy and fast computational speed is steadily increasing. To address this challenge, we present an efficient and accurate algorithm for high-dimensional numerical modeling. This approach combines a central finite-difference method with the discrete Sine transform (DST) scheme to solve the Poisson equation under Dirichlet boundary conditions (DBCs). To balance numerical accuracy and computation, the DST scheme is applied along one direction in the 2D case and two directions in the 3D case. This strategy effectively reduces problem complexity while maintaining low computational cost. The hybrid DST-accelerated finite-difference approach substantially lowers the computational cost associated with solving the Poisson equation on large grids. Comprehensive numerical experiments for 2D and 3D Poisson equations with DBCs have been conducted. The obtained numerical results demonstrate that the proposed hybrid method not only significantly reduces the computational expenses, but also maintains the central finite-difference accuracy. Full article
(This article belongs to the Special Issue Numerical Methods for Scientific Computing)
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14 pages, 324 KB  
Article
An Inexact Noda Iteration for Computing the Smallest Eigenpair of a Large, Irreducible Monotone Matrix
by Ching-Sung Liu
Mathematics 2024, 12(16), 2546; https://doi.org/10.3390/math12162546 - 17 Aug 2024
Viewed by 1645
Abstract
In this paper, we introduce an inexact Noda iteration method featuring inner and outer iterations for computing the smallest eigenvalue and corresponding eigenvector of an irreducible monotone matrix. The proposed method includes two primary relaxation steps designed to compute the smallest eigenvalue and [...] Read more.
In this paper, we introduce an inexact Noda iteration method featuring inner and outer iterations for computing the smallest eigenvalue and corresponding eigenvector of an irreducible monotone matrix. The proposed method includes two primary relaxation steps designed to compute the smallest eigenvalue and its associated eigenvector. These steps are influenced by specific relaxation factors, and we examine how these factors impact the convergence of the outer iterations. By applying two distinct relaxation factors to solve the inner linear systems, we demonstrate that the convergence can be globally linear or superlinear, contingent upon the relaxation factor used. Additionally, the relaxation factor affects the rate of convergence. The inexact Noda iterations we propose are structure-preserving and ensure the positivity of the approximate eigenvectors. Numerical examples are provided to demonstrate the practicality of the proposed method, consistently preserving the positivity of approximate eigenvectors. Full article
(This article belongs to the Special Issue Numerical Methods for Scientific Computing)
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