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Mathematics
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High-Order Numerical Methods and Computational Fluid Dynamics
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High-Order Numerical Methods and Computational Fluid Dynamics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: 30 November 2026 | Viewed by 3253

Special Issue Editor

Dr. KyungSung Kim

E-Mail Website
Guest Editor
Department of Mechatronics Convergence Engineering, Changwon National University, Changwon, Republic of Korea
Interests: particle-based simulation; high-order numerical applications; fluid–structure interaction

Special Issue Information

Dear Colleagues,

In this Special Issue, we aim to present the recent developments in the theory and applications of high-order numerical methods and computational fluid dynamics including conventional and particle applications. This special issue will accept high-quality papers containing original research results and review articles of exceptional merit in the following fields: particle-based simulation, high-order multiphase instability, hydrodynamics, computational fluid dynamics with Neural networks.

Dr. KyungSung Kim
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • high-order numerical method
  • computational fluid dynamics
  • particle-based simulation method
  • neural network in CFD
  • numerical schemes and algorithms
  • high-order hydrodynamics

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Published Papers (4 papers)

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Research

39 pages, 6563 KB  
Article
Model Predictive Control for Dynamic Positioning of a Fireboat Considering Non-Linear Environmental Disturbances and Water Cannon Reaction Forces Based on Numerical Modeling
by Dabin Lee and Sewon Kim
Mathematics 2026, 14(3), 401; https://doi.org/10.3390/math14030401 - 23 Jan 2026
Viewed by 535
Abstract
Dynamic positioning (DP) systems play a critical role in maintaining vessel position and heading under environmental disturbances such as wind, waves, and currents. This study presents a model predictive control (MPC)-based DP system for a fireboat equipped with a rudder–propeller configuration, explicitly accounting [...] Read more.
Dynamic positioning (DP) systems play a critical role in maintaining vessel position and heading under environmental disturbances such as wind, waves, and currents. This study presents a model predictive control (MPC)-based DP system for a fireboat equipped with a rudder–propeller configuration, explicitly accounting for both environmental loads and the reaction force generated during water cannon operation. Unlike conventional DP architectures in which DP control and thrust allocation are treated as separate modules, the proposed framework integrates both functions within a unified MPC formulation, enabling real-time optimization under actuator constraints. Environmental loads are modeled by incorporating nonlinear second-order wave drift effects, while nonlinear rudder–propeller interaction forces are derived through computational fluid dynamics (CFD) analysis and embedded in a control-oriented dynamic model. This modeling approach allows operational constraints, including rudder angle limits and propeller thrust saturation, to be explicitly considered in the control formulation. Simulation results demonstrate that the proposed MPC-based DP system achieves improved station-keeping accuracy, enhanced stability, and increased robustness against combined environmental disturbances and water cannon reaction forces, compared to a conventional PID controller. Full article
(This article belongs to the Special Issue High-Order Numerical Methods and Computational Fluid Dynamics)
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36 pages, 42027 KB  
Article
DStreaM: A Convective Term Approximation Approach That Corresponds to Pure Convection
by Kiril Shterev
Mathematics 2026, 14(3), 389; https://doi.org/10.3390/math14030389 - 23 Jan 2026
Viewed by 333
Abstract
In recent decades, considerable effort has been devoted to developing higher-order schemes for the discretization of convective terms that are both stable and reliable. In this work, the central idea is that the approximation should be made to reflect the physics of pure [...] Read more.
In recent decades, considerable effort has been devoted to developing higher-order schemes for the discretization of convective terms that are both stable and reliable. In this work, the central idea is that the approximation should be made to reflect the physics of pure convection: the transported quantity is advected along streamlines, and information is propagated only in the upwind direction, i.e., the transported property is determined by previous values along the streamline but not by downstream values. In the proposed approach, streamlines on the computational mesh are represented by discrete streamlines, and the method is called the Discrete Streamline Method (DStreaM). A discrete streamline is constructed as a narrow triangle with one vertex at the node where the approximation is sought and two vertices at upstream neighbouring nodes. Discrete streamlines are oriented according to the local flow direction, in a manner similar to skew-upwind schemes, so that consistency with pure convection is ensured for DStreaM. The method is conservative only for uniform meshes with a constant velocity field; for general meshes and non-uniform velocity fields, it is non-conservative, and a non-zero local conservation error remains. The performance of DStreaM is assessed on the following standard test problems: convection of a step profile, a double-step profile, a sinusoidal profile, and the Smith–Hutton problem. DStreaM solutions are compared with those obtained using the first-order upwind scheme and second-order total variation diminishing (TVD) schemes with Minmod, QUICK, and SUPERBEE limiters. Across these benchmarks, high-resolution solution profiles and L1/L2 error levels comparable to those of the considered TVD schemes are produced by DStreaM. In the DStreaM construction, only local node coordinates and mesh connectivity are used; in this work, implementation is performed on both uniform Cartesian meshes and unstructured triangular meshes generated by a Delaunay triangulation. Representative results are reported with a focus on accuracy, iterative convergence, and conservation limitations. Full article
(This article belongs to the Special Issue High-Order Numerical Methods and Computational Fluid Dynamics)
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17 pages, 12479 KB  
Article
A Study of Sediment Behavior for Dam-Break Flow over Granular Bed
by Kyung Sung Kim
Mathematics 2025, 13(24), 3919; https://doi.org/10.3390/math13243919 - 8 Dec 2025
Cited by 1 | Viewed by 615
Abstract
Dam-break flows involve strong non-linearity and complex fluid–solid interactions, often causing severe flooding and structural damage. Particle-based CFD methods, such as the Moving Particle Semi-implicit (MPS) method, are effective in modeling such flows due to their mesh-free, Lagrangian nature. This study presents an [...] Read more.
Dam-break flows involve strong non-linearity and complex fluid–solid interactions, often causing severe flooding and structural damage. Particle-based CFD methods, such as the Moving Particle Semi-implicit (MPS) method, are effective in modeling such flows due to their mesh-free, Lagrangian nature. This study presents an improved MPS method with a novel friction model and enhanced fluid–solid interaction scheme to simulate dam-break-induced flows over fixed and mobile beds. The model is validated using experimental and analytical benchmarks, demonstrating improved accuracy and stability. Simulation results show that mobile beds significantly influence wave attenuation, energy dissipation, and sediment transport. In particular, step-down bed conditions promote sediment motion and modify wave behavior. These findings emphasize the importance of accounting for mobile seabed dynamics in numerical modeling of coastal and dam-break scenarios. The proposed MPS model offers a reliable and efficient tool for capturing key phenomena associated with fluid–solid interactions in naval and ocean engineering applications. Full article
(This article belongs to the Special Issue High-Order Numerical Methods and Computational Fluid Dynamics)
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26 pages, 43661 KB  
Article
Numerical Investigation of Atwood Number Effects on Shock-Driven Single-Mode Stratified Heavy Fluid Layers
by Salman Saud Alsaeed, Satyvir Singh and Nouf A. Alrubea
Mathematics 2025, 13(18), 3032; https://doi.org/10.3390/math13183032 - 19 Sep 2025
Cited by 5 | Viewed by 976
Abstract
This work presents a numerical investigation of Richtmyer–Meshkov instability (RMI) in shock-driven single-mode stratified heavy fluid layers, with emphasis on the influence of the Atwood number. High-order modal discontinuous Galerkin simulations are carried out for Atwood numbers ranging from A=0.30 to [...] Read more.
This work presents a numerical investigation of Richtmyer–Meshkov instability (RMI) in shock-driven single-mode stratified heavy fluid layers, with emphasis on the influence of the Atwood number. High-order modal discontinuous Galerkin simulations are carried out for Atwood numbers ranging from A=0.30 to 0.72, allowing a systematic study of interface evolution, vorticity dynamics, and mixing. The analysis considers diagnostic quantities such as interface trajectories, normalized interface length and amplitude, vorticity extrema, circulation, enstrophy, and kinetic energy. The results demonstrate that the Atwood number plays a central role in instability development. At low A, interface deformation remains smooth and coherent, with weaker vorticity deposition and delayed nonlinear roll-up. As A increases, baroclinic torque intensifies, leading to rapid perturbation growth, stronger vortex roll-ups, and earlier onset of secondary instabilities such as Kelvin–Helmholtz vortices. Enstrophy, circulation, and interface measures show systematic amplification with increasing density contrast, while the total kinetic energy exhibits relatively weak sensitivity to A. Overall, the study highlights how the Atwood number governs the transition from linear to nonlinear dynamics, controlling both large-scale interface morphology and the formation of small-scale vortical structures. These findings provide physical insight into shock–interface interactions and contribute to predictive modeling of instability-driven mixing in multicomponent flows. Full article
(This article belongs to the Special Issue High-Order Numerical Methods and Computational Fluid Dynamics)
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