Article
Computer Science, Interdisciplinary Applications
Nat H. Mathews, Natasha Flyer, Sarah E. Gibson
Summary: The study presents a novel magnetohydrostatic numerical model for directly solving the force-balanced magnetic field in the solar corona. The model utilizes Radial Basis Function Finite Differences with 3D polyharmonic splines plus polynomials as the core discretization. It addresses the challenges posed by the ill-posed and numerically intractable nature of the static force-balance equations in the limit of zero forcing.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Engineering, Multidisciplinary
Kyle W. Beggs, Eduardo Divo, Alain J. Kassab
Summary: In this paper, a localized Radial-Basis Function collocation Meshless flow solver is developed and tightly coupled to a 0D Lumped-Parameter Model for accurate hemodynamic simulations. This approach is well-suited for modeling complex non-Newtonian hemodynamics and allows for fast computations.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Mathematics
Malik Zaka Ullah, Abdullah Khamis Alzahrani, Hashim Mohammed Alshehri, Stanford Shateyi
Summary: In this work, a numerical scheme is presented to solve the time-fractional Black-Scholes PDE under the generalized multiquadric radial basis function. The scheme uses spatial uniform meshes and stencils with five adjacent discretization nodes and estimates the time-fractional derivative using an L1-scheme. The results of numerical tests demonstrate the efficacy of the presented solver.
Article
Mathematics, Applied
Jie Hou, Ying Li, Shihui Ying
Summary: In this paper, a novel iterative optimization method is proposed to determine the best parameter c for the Radial basis function finite difference (RBF-FD) method based on the Double Operator Error (DOE). This method is a general iterative optimization approach that can rapidly determine the optimal c for a given problem and significantly reduce the numerical error for any type of radial basis functions (RBFs). Three numerical examples demonstrate the effectiveness and generality of the proposed method compared to previous methods. The relevant data and code can be found at https://github.com/hsbhc/IQM-RBF-FD.& COPY; 2023 Published by Elsevier Ltd.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Sara Arefian, Davoud Mirzaei
Summary: In this work, the standard Hermite interpolation based RBF-HFD method is developed into a new faster and more accurate technique based on the PU method. The new approach solves much fewer local linear systems for calculating stencil weights, reducing computational cost. The method also allows flexibility in using different types of PU weight functions and utilizes the scaling property of PHS kernels.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
Reyhaneh Mir, Davoud Mirzaei
Summary: In this paper, a new direct RBF partition of unity (D-RBF-PU) method is developed for numerical solution of partial differential equations defined on smooth orientable surfaces discretized with scattered nodes and approximations to normal vectors. The accuracy, stability, and efficiency of the method are studied through theoretical and experimental results. The method proves to be superior to other comparable techniques for surface PDEs.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Engineering, Multidisciplinary
Victor Bayona, Mario Sanchez-Sanz, Eduardo Fernandez-Tarrazo, Manuel Kindelan
Summary: This research focuses on developing a high-order meshfree method to model combustion inside complex geometries using radial basis functions-generated finite differences, aiming to identify different combustion regimes and improve conversion efficiency.
APPLIED MATHEMATICAL MODELLING
(2021)
Article
Mathematics, Applied
Davoud Mirzaei
Summary: This paper proposes a new localized radial basis function method based on partition of unity for solving boundary and initial-boundary value problems. The new method, called the direct RBF partition of unity (D-RBF-PU) method, is faster and simpler than the standard RBF-PU method by avoiding derivatives of PU weight functions and lower derivatives of local approximants. Additionally, the method is more efficient and less expensive with the use of discontinuous PU weight functions, and numerical experiments on irregular domains support the efficiency of the new method.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Engineering, Multidisciplinary
Faranak Gholampour, Esmail Hesameddini, Ameneh Taleei
Summary: This work presents a numerical solution for two-dimensional elasticity problems involving multiple material phases. The proposed method, based on radial basis functions, has been proven to be effective in solving partial differential equations on irregular domains. By utilizing polynomial augmented polyharmonic spline radial basis functions, the method provides highly accurate results while bypassing stability issues and stagnation errors.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2022)
Article
Computer Science, Interdisciplinary Applications
Varun Shankar, Grady B. Wright, Aaron L. Fogelson
Summary: The study introduces a high-order radial basis function finite difference method for solving advection-diffusion equations on time-varying domains. The framework eliminates overlap parameters, enables tuning-free assembly of differentiation matrices on moving domains, and demonstrates high performance with high convergence rates.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Andrew M. Jones, Peter A. Bosler, Paul A. Kuberry, Grady B. Wright
Summary: Approximating differential operators on two-dimensional surfaces is a crucial problem in various fields. Localized meshfree methods, such as generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD), have been proven effective and efficient in achieving high accuracy at low computational cost for this task. However, a direct comparison of these methods for approximating surface differential operators (SDOs) has not been conducted yet. This study aims to fill this gap and compare the performance of GMLS with an RBF-FD method based on polyharmonic spline kernels and polynomials (PHS+Poly). Furthermore, we investigate the relationship between the tangent plane formulation of SDOs and the local coordinate formulation used in GMLS, and propose a new RBF-FD method for approximating the tangent space of an unknown point cloud surface using ideas from the GMLS SDO formulation. Evaluation: 8/10.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Engineering, Multidisciplinary
M. N. Rasoulizadeh, M. J. Ebadi, Z. Avazzadeh, O. Nikan
Summary: The paper presents an accurate and robust meshless approach for solving the nonlinear equal width equation, using localized radial basis function-finite difference method and implicit techniques. The stability analysis and comparison with other techniques are also conducted to assess the validity, efficiency, and accuracy of the method.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2021)
Article
Mathematics, Applied
Grady B. Wright, Andrew Jones, Varun Shankar
Summary: We propose a new meshfree geometric multilevel (MGM) method for solving linear systems arising from discretizing elliptic PDEs on point cloud surfaces. The method utilizes Poisson disk sampling for coarsening point clouds and uses polyharmonic splines for transferring information. It is applicable to various localized meshfree methods and has been tested on different problems, showing efficient convergence rates and scalability. The effectiveness of MGM is further demonstrated on challenging applications involving complicated surfaces.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Bengt Fornberg
Summary: The trapezoidal rule is accurate for integrals over periodic intervals but inaccurate in nonperiodic cases. Common improvements, such as Simpson's rule and the Newton-Cotes formulas, may not be better than classical quadrature formulas. These methods suffer from the Runge phenomenon for increasing orders of accuracy.
Article
Mathematics
Tony Liu, Rodrigo B. Platte
Summary: Polyharmonic spline (PHS) radial basis functions (RBFs) are often used in RBF finite-difference (RBF-FD) methods alongside polynomials. Novel strategies for computing the placement of sampling points in both 1D and 2D have been explored, determining the optimality of sampling points through a piecewise-defined Lebesgue constant. By modifying a column-pivoting QR algorithm, sampling points can be selected to reduce computational costs and maintain accuracy in RBF-FD methods.
Article
Computer Science, Interdisciplinary Applications
Sebastien Blaise, Jonathan Lambrechts, Eric Deleersnijder
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2016)
Article
Mathematics, Applied
Edward J. Fuselier, Grady B. Wright
IMA JOURNAL OF NUMERICAL ANALYSIS
(2017)
Article
Computer Science, Interdisciplinary Applications
Grady B. Wright, Bengt Fornberg
JOURNAL OF COMPUTATIONAL PHYSICS
(2017)
Article
Mathematics, Applied
Heather Wilber, Alex Townsend, Grady B. Wright
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2017)
Article
Mathematics, Applied
Erik Lehto, Varun Shankar, Grady B. Wright
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2017)
Article
Computer Science, Interdisciplinary Applications
Varun Shankar, Grady B. Wright
JOURNAL OF COMPUTATIONAL PHYSICS
(2018)
Article
Computer Science, Interdisciplinary Applications
Kathryn P. Drake, Grady B. Wright
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Computer Science, Interdisciplinary Applications
Kathryn P. Drake, Grady B. Wright
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Computer Science, Interdisciplinary Applications
Sebastien Blaise, Benoit Spinewine
Summary: The article introduces a new method for optimizing the least cost route, which incorporates curvature constraints into primary calculations, eliminating the need for post-process smoothing and preserving the optimal character of the route. By adapting optimization algorithms for forward-moving vehicles, faster and more accurate results are achieved. This method offers higher flexibility in local route orientation compared to traditional algorithms.
ENGINEERING WITH COMPUTERS
(2022)
Article
Computer Science, Interdisciplinary Applications
Varun Shankar, Grady B. Wright, Aaron L. Fogelson
Summary: The study introduces a high-order radial basis function finite difference method for solving advection-diffusion equations on time-varying domains. The framework eliminates overlap parameters, enables tuning-free assembly of differentiation matrices on moving domains, and demonstrates high performance with high convergence rates.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Kathryn P. Drake, Edward J. Fuselier, Grady B. Wright
Summary: This paper presents a new method for surface reconstruction from a point cloud by utilizing the approximate normals to the surface. By using curl-free radial basis function interpolation of the normals, an implicit surface approximation for the point cloud can be obtained. The method is combined with a partition of unity technique to better represent local features and handle noise in both the normals and the point positions.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Grady B. Wright, Andrew Jones, Varun Shankar
Summary: We propose a new meshfree geometric multilevel (MGM) method for solving linear systems arising from discretizing elliptic PDEs on point cloud surfaces. The method utilizes Poisson disk sampling for coarsening point clouds and uses polyharmonic splines for transferring information. It is applicable to various localized meshfree methods and has been tested on different problems, showing efficient convergence rates and scalability. The effectiveness of MGM is further demonstrated on challenging applications involving complicated surfaces.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Andrew M. Jones, Peter A. Bosler, Paul A. Kuberry, Grady B. Wright
Summary: Approximating differential operators on two-dimensional surfaces is a crucial problem in various fields. Localized meshfree methods, such as generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD), have been proven effective and efficient in achieving high accuracy at low computational cost for this task. However, a direct comparison of these methods for approximating surface differential operators (SDOs) has not been conducted yet. This study aims to fill this gap and compare the performance of GMLS with an RBF-FD method based on polyharmonic spline kernels and polynomials (PHS+Poly). Furthermore, we investigate the relationship between the tangent plane formulation of SDOs and the local coordinate formulation used in GMLS, and propose a new RBF-FD method for approximating the tangent space of an unknown point cloud surface using ideas from the GMLS SDO formulation. Evaluation: 8/10.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Kathryn P. Drake, Edward J. Fuselier, Grady B. Wright
Summary: The paper introduces a technique for constructing global approximants of divergence-free or curl-free vector fields by combining div/curl-free radial basis functions in a partition of unity framework, applicable to vector fields in 2D space and on surfaces, and providing approximations for scalar potentials. The method effectively bypasses the computational expense issue caused by the global nature of the problem.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Varun Shankar, Grady B. Wright, Akil Narayan
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2020)
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Lin Fu
Summary: In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Lukas Lundgren, Murtazo Nazarov
Summary: In this paper, a high-order accurate finite element method for incompressible variable density flow is introduced. The method addresses the issues of saddle point system and stability problem through Schur complement preconditioning and artificial compressibility approaches, and it is validated to have high-order accuracy for smooth problems and accurately resolve discontinuities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Gabriele Ciaramella, Laurence Halpern, Luca Mechelli
Summary: This paper presents a novel convergence analysis of the optimized Schwarz waveform relaxation method for solving optimal control problems governed by periodic parabolic PDEs. The analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition, which enables a precise characterization of the convergence factor at the semidiscrete level. The behavior of the optimal transmission condition parameter is also analyzed in detail as the time discretization approaches zero.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jonas A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Summary: This article introduces a scientific machine learning framework that uses a partition of unity architecture to model physics through control volume analysis. The framework can extract reduced models from full field data while preserving the physics. It is applicable to manifolds in arbitrary dimension and has been demonstrated effective in specific problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Nozomi Magome, Naoki Morita, Shigeki Kaneko, Naoto Mitsume
Summary: This paper proposes a novel strategy called B-spline based SFEM to fundamentally solve the problems of the conventional SFEM. It uses different basis functions and cubic B-spline basis functions with C-2-continuity to improve the accuracy of numerical integration and avoid matrix singularity. Numerical results show that the proposed method is superior to conventional methods in terms of accuracy and convergence.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Timothy R. Law, Philip T. Barton
Summary: This paper presents a practical cell-centred volume-of-fluid method for simulating compressible solid-fluid problems within a pure Eulerian setting. The method incorporates a mixed-cell update to maintain sharp interfaces, and can be easily extended to include other coupled physics. Various challenging test problems are used to validate the method, and its robustness and application in a multi-physics context are demonstrated.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Xing Ji, Fengxiang Zhao, Wei Shyy, Kun Xu
Summary: This paper presents the development of a third-order compact gas-kinetic scheme for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme demonstrates robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, Arunasalam Rahunanthan
Summary: This paper presents a novel Bayesian statistical framework for the characterization of natural subsurface formations, and introduces the concept of multiscale sampling to localize the search in the stochastic space. The results show that the proposed framework performs well in solving inverse problems related to porous media flows.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jacob Rains, Yi Wang, Alec House, Andrew L. Kaminsky, Nathan A. Tison, Vamshi M. Korivi
Summary: This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Junhong Yue, Peijun Li
Summary: This paper proposes two numerical methods (IP-FEM and BP-FEM) to study the flexural wave scattering problem of an arbitrary-shaped cavity on an infinite thin plate. These methods successfully decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary, providing an effective solution to this challenging problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
William Anderson, Mohammad Farazmand
Summary: We develop fast and scalable methods, called RONS, for computing reduced-order nonlinear solutions. These methods have been proven to be highly effective in tackling challenging problems, but become computationally prohibitive as the number of parameters grows. To address this issue, three separate methods are proposed and their efficacy is demonstrated through examples. The application of RONS to neural networks is also discussed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini
Summary: In this paper, a second order exponential scheme for stiff evolutionary advection-diffusion-reaction equations is proposed. The scheme is based on a directional splitting approach and uses computation of small sized exponential-like functions and tensor-matrix products for efficient implementation. Numerical examples demonstrate the advantage of the proposed approach over state-of-the-art techniques.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo
Summary: This work proposes a high order conservative semi-Lagrangian method for the inhomogeneous Boltzmann equation of rarefied gas dynamics. The method combines a semi-Lagrangian scheme for the convection term, a fast spectral method for computation of the collision operator, and a high order conservative reconstruction and a weighted optimization technique to preserve conservative quantities. Numerical tests demonstrate the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jialei Li, Xiaodong Liu, Qingxiang Shi
Summary: This study shows that the number, centers, scattering strengths, inner and outer diameters of spherical shell-structured sources can be uniquely determined from the far field patterns. A numerical scheme is proposed for reconstructing the spherical shell-structured sources, which includes a migration series method for locating the centers and an iterative method for computing the inner and outer diameters without computing derivatives.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
