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Open Source Codes for Numerical Analysis
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Open Source Codes for Numerical Analysis

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 28424

Special Issue Editors

Prof. Dr. Junseok Kim

E-Mail Website
Guest Editor
Department of Mathematics, Korea University, Seoul, Korea
Interests: computational mathematics; scientific computing; numerical analysis; mathematical physics; computational biology; computational finance; computer simulation; mathematical modeling
Prof. Dr. Darae Jeong

E-Mail Website
Guest Editor
Department of Mathematics, Kangwon National University, Gangwon-do, Chuncheon 200-701, Korea
Interests: scientific computation; numerical analysis; computational physics; computational finance; computational biology
Prof. Dr. Yongho Choi

E-Mail Website
Guest Editor
Department of Mathematics and Big Data, Daegu University, 201 Daegudae-ro Gyeongsangbuk-do, Gyeongsan 38453, Korea
Interests: PDE on surfaces; pattern formation; mathematical biology; image analysis; industrial mathematics

Special Issue Information

Dear Colleagues,

Computational mathematics is growing rapidly with high computing power. Many researchers have developed their own computer codes for the mathematical models of various important problems. Nowadays, interdisciplinary studies are becoming more and more popular. Therefore, there is a need to incorporate other research areas into their own expertise. However, it is very difficult to implement computer codes for beginners in a new research area. In this Special Issue, we invite papers which describe a brief review of numerical methods, give detailed explanations of source code, and provide the source code itself so that readers can easily use the provided code and modify the code to suit their needs.

Prof. Dr. Junseok Kim
Prof. Dr. Darae Jeong
Prof. Dr. Yongho Choi
Guest Editors

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 100 words) can be sent to the Editorial Office for announcement on this website.

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

  • Phase-field method
  • Level-set method
  • Immersed boundary method
  • Lattice Boltzmann method
  • Multigrid method
  • Fourier Spectral method for PDE
  • Operator splitting method
  • Topological optimization
  • Mesh generation algorithm
  • Boundary integral method
  • PDE solver on curved surfaces

Published Papers (7 papers)

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Research

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20 pages, 4105 KiB  
Article
Numerical Investigation of Freely Falling Objects Using Direct-Forcing Immersed Boundary Method
by Cheng-Shu You, Ming-Jyh Chern, Dedy Zulhidayat Noor and Tzyy-Leng Horng
Mathematics 2020, 8(9), 1619; https://0-doi-org.brum.beds.ac.uk/10.3390/math8091619 - 18 Sep 2020
Cited by 3 | Viewed by 3183
Abstract
The fluid-structure interaction of solid objects freely falling in a Newtonian fluid was investigated numerically by direct-forcing immersed boundary (DFIB) method. The Navier–Stokes equations are coupled with equations of motion through virtual force to describe the motion of solid objects. Here, we rigorously [...] Read more.
The fluid-structure interaction of solid objects freely falling in a Newtonian fluid was investigated numerically by direct-forcing immersed boundary (DFIB) method. The Navier–Stokes equations are coupled with equations of motion through virtual force to describe the motion of solid objects. Here, we rigorously derived the equations of motion by taking control-volume integration of momentum equation. The method was validated by a popular numerical test example describing the 2D flow caused by the free fall of a circular disk inside a tank of fluid, as well as 3D experimental measurements in the sedimentation of a sphere. Then, we demonstrated the method by a few more 2D sedimentation examples: (1) free fall of two tandem circular disks showing drafting, kissing and tumbling phenomena; (2) sedimentation of multiple circular disks; (3) free fall of a regular triangle, in which the rotation of solid object is significant; (4) free fall of a dropping ellipse to mimic the falling of a leaf. In the last example, we found rich falling patterns exhibiting fluttering, tumbling, and chaotic falling. Full article
(This article belongs to the Special Issue Open Source Codes for Numerical Analysis)
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10 pages, 2954 KiB  
Article
An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation
by Hyun Geun Lee
Mathematics 2020, 8(9), 1502; https://0-doi-org.brum.beds.ac.uk/10.3390/math8091502 - 04 Sep 2020
Cited by 5 | Viewed by 1898
Abstract
The conservative Swift–Hohenberg equation was introduced to reformulate the phase-field crystal model. A challenge in solving the conservative Swift–Hohenberg equation numerically is how to treat the nonlinear term to preserve mass conservation without compromising efficiency and accuracy. To resolve this problem, we present [...] Read more.
The conservative Swift–Hohenberg equation was introduced to reformulate the phase-field crystal model. A challenge in solving the conservative Swift–Hohenberg equation numerically is how to treat the nonlinear term to preserve mass conservation without compromising efficiency and accuracy. To resolve this problem, we present a linear, high-order, and mass conservative method by placing the linear and nonlinear terms in the implicit and explicit parts, respectively, and employing the implicit-explicit Runge–Kutta method. We show analytically that the method inherits the mass conservation. Numerical experiments are presented demonstrating the efficiency and accuracy of the proposed method. In particular, long time simulation for pattern formation in 2D is carried out, where the phase diagram can be observed clearly. The MATLAB code for numerical implementation of the proposed method is provided in Appendix. Full article
(This article belongs to the Special Issue Open Source Codes for Numerical Analysis)
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14 pages, 587 KiB  
Article
A Simple Parallel Solution Method for the Navier–Stokes Cahn–Hilliard Equations
by Nadja Adam, Florian Franke and Sebastian Aland
Mathematics 2020, 8(8), 1224; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081224 - 25 Jul 2020
Cited by 8 | Viewed by 4196
Abstract
We present a discretization method of the Navier–Stokes Cahn–Hilliard equations which offers an impressing simplicity, making it easy to implement a scalable parallel code from scratch. The method is based on a special pressure projection scheme with incomplete pressure iterations. The resulting scheme [...] Read more.
We present a discretization method of the Navier–Stokes Cahn–Hilliard equations which offers an impressing simplicity, making it easy to implement a scalable parallel code from scratch. The method is based on a special pressure projection scheme with incomplete pressure iterations. The resulting scheme admits solution by an explicit Euler method. Hence, all unknowns decouple, which enables a very simple implementation. This goes along with the opportunity of a straightforward parallelization, for example, by few lines of Open Multi-Processing (OpenMP) or Message Passing Interface (MPI) routines. Using a standard benchmark case of a rising bubble, we show that the method provides accurate results and good parallel scalability. Full article
(This article belongs to the Special Issue Open Source Codes for Numerical Analysis)
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Review

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16 pages, 6343 KiB  
Review
Immersed Boundary Method for Simulating Interfacial Problems
by Wanho Lee and Seunggyu Lee
Mathematics 2020, 8(11), 1982; https://0-doi-org.brum.beds.ac.uk/10.3390/math8111982 - 06 Nov 2020
Viewed by 2177
Abstract
We review the immersed boundary (IB) method in order to investigate the fluid-structure interaction problems governed by the Navier–Stokes equation. The configuration is described by the Lagrangian variables, and the velocity and pressure of the fluid are defined in Cartesian coordinates. The interaction [...] Read more.
We review the immersed boundary (IB) method in order to investigate the fluid-structure interaction problems governed by the Navier–Stokes equation. The configuration is described by the Lagrangian variables, and the velocity and pressure of the fluid are defined in Cartesian coordinates. The interaction between two different coordinates is involved in a discrete Dirac-delta function. We describe the IB method and its numerical implementation. Standard numerical simulations are performed in order to show the effect of the parameters and discrete Dirac-delta functions. Simulations of flow around a cylinder and movement of Caenorhabditis elegans are introduced as rigid and flexible boundary problems, respectively. Furthermore, we provide the MATLAB codes for our simulation. Full article
(This article belongs to the Special Issue Open Source Codes for Numerical Analysis)
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11 pages, 304 KiB  
Review
Finite Difference Method for the Hull–White Partial Differential Equations
by Yongwoong Lee and Kisung Yang
Mathematics 2020, 8(10), 1719; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101719 - 07 Oct 2020
Cited by 1 | Viewed by 3925
Abstract
This paper reviews the finite difference method (FDM) for pricing interest rate derivatives (IRDs) under the Hull–White Extended Vasicek model (HW model) and provides the MATLAB codes for it. Among the financial derivatives on various underlying assets, IRDs have the largest trading volume [...] Read more.
This paper reviews the finite difference method (FDM) for pricing interest rate derivatives (IRDs) under the Hull–White Extended Vasicek model (HW model) and provides the MATLAB codes for it. Among the financial derivatives on various underlying assets, IRDs have the largest trading volume and the HW model is widely used for pricing them. We introduce general backgrounds of the HW model, its associated partial differential equations (PDEs), and FDM formulation for one- and two-asset problems. The two-asset problem is solved by the basic operator splitting method. For numerical tests, one- and two-asset bond options are considered. The computational results show close values to analytic solutions. We conclude with a brief comment on the research topics for the PDE approach to IRD pricing. Full article
(This article belongs to the Special Issue Open Source Codes for Numerical Analysis)
36 pages, 12863 KiB  
Review
Fourier-Spectral Method for the Phase-Field Equations
by Sungha Yoon, Darae Jeong, Chaeyoung Lee, Hyundong Kim, Sangkwon Kim, Hyun Geun Lee and Junseok Kim
Mathematics 2020, 8(8), 1385; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081385 - 18 Aug 2020
Cited by 22 | Viewed by 6331
Abstract
In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific [...] Read more.
In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A. Full article
(This article belongs to the Special Issue Open Source Codes for Numerical Analysis)
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17 pages, 514 KiB  
Review
Finite Difference Method for the Multi-Asset Black–Scholes Equations
by Sangkwon Kim, Darae Jeong, Chaeyoung Lee and Junseok Kim
Mathematics 2020, 8(3), 391; https://0-doi-org.brum.beds.ac.uk/10.3390/math8030391 - 10 Mar 2020
Cited by 7 | Viewed by 5488
Abstract
In this paper, we briefly review the finite difference method (FDM) for the Black–Scholes (BS) equations for pricing derivative securities and provide the MATLAB codes in the Appendix for the one-, two-, and three-dimensional numerical implementation. The BS equation is discretized non-uniformly in [...] Read more.
In this paper, we briefly review the finite difference method (FDM) for the Black–Scholes (BS) equations for pricing derivative securities and provide the MATLAB codes in the Appendix for the one-, two-, and three-dimensional numerical implementation. The BS equation is discretized non-uniformly in space and implicitly in time. The two- and three-dimensional equations are solved using the operator splitting method. In the numerical tests, we show characteristic examples for option pricing. The computational results are in good agreement with the closed-form solutions to the BS equations. Full article
(This article belongs to the Special Issue Open Source Codes for Numerical Analysis)
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