Article
Computer Science, Interdisciplinary Applications
Zelalem Arega Worku, David W. Zingg
Summary: This study analyzes several types of SAT for diffusion problems discretized with diagonal-norm multidimensional SBP operators, presenting conditions for consistency, conservation, adjoint consistency, and energy stability. The theoretical results are supported by numerical experiments conducted on the two-dimensional Poisson problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics
Artur Karimov, Denis Butusov, Valery Andreev, Erivelton G. Nepomuceno
Summary: This paper introduces a numerical-analytical method based on rational approximation, which outperforms the Taylor series and Runge-Kutta methods in solving highly stiff problems of the same accuracy order. The method efficiently addresses issues of order, stability, and adaptive step control.
Article
Computer Science, Interdisciplinary Applications
Alexander Rothkopf, Jan Nordstrom
Summary: Motivated by the connection between classical and quantum physics, this study presents a new discretization prescription for classical initial value problems (IVPs). By formulating the problem as an optimization task, classical trajectories can be obtained without deriving an equation of motion. The demonstrated numerical experiments show the stability, accuracy, and convergence properties of this approach in systems with classical equations of motion.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Ahmad Deeb, Aziz Hamdouni, Dina Razafindralandy
Summary: A stability analysis is conducted on the Borel-Pade-Laplace series summation technique, used as an explicit time integrator, and its numerical performance on stiff and non-stiff problems is analyzed. Applications of this technique to ordinary and partial differential equations are presented, and the results are compared with other popular schemes designed for stiff and non-stiff equations.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Interdisciplinary Applications
Hendrik Ranocha
Summary: Nishikawa (2007) proposed a reformulation of the classical Poisson equation as a steady state problem for a linear hyperbolic system, which provides optimal error estimates for the solution of the elliptic equation and its gradient. However, it hinders the use of well-known solvers for elliptic problems. We establish connections to a discontinuous Galerkin (DG) method studied by Cockburn, Guzman, and Wang (2009) that is generally difficult to implement. Additionally, we demonstrate the efficient implementation of this method using summation by parts (SBP) operators, particularly in the context of SBP DG methods like the DG spectral element method (DGSEM). The resulting scheme combines desirable properties from both the hyperbolic and the elliptic perspective, offering a higher order of convergence for the gradients than what is typically expected from DG methods for elliptic problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Jan Nordstrom
Summary: Linearisation is used to analyze nonlinear initial boundary value problems by first converting them into linear problems. This paper attempts to resolve the contradiction between energy conservation and boundedness in the nonlinear and linearised versions of the problem. The authors propose a specific skew-symmetric form and a non-standard linearisation procedure that preserve these properties in the linearised problem and its related dual problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Jan Glaubitz, Simon-Christian Klein, Jan Nordstroem, Philipp Oeffner
Summary: Summation-by-parts (SBP) operators are used to develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. The existing SBP operators are based on the idea that polynomials can accurately approximate the solution, but this may not always be the best approximation. Function-space SBP (FSBP) operators are developed to address this issue. In this paper, the theory of FSBP operators is extended to multiple dimensions, and it is shown that the concept of SBP operators can be applied to a significantly larger class of methods.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Viktor Linders
Summary: This paper presents three results regarding the eigenvalue property of SBP operators. It shows that not all nullspace consistent SBP operators possess this property, but it can be addressed by adding a perturbation term without affecting accuracy. Additionally, it proves that all pseudospectral methods satisfy the eigenvalue property.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
A. G. Fareo
Summary: The paper provides a concise review on using the scaling transformation group to determine a modified boundary value problem, which is invariant under an extended two-parameter scaling group. The main contribution is the development of a simple iterative method that does not require the extended group.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics
Saisai Yang, Tusheng Zhang
Summary: In this paper, a unique weak solution to the Dirichlet boundary value problem for second order elliptic operators with coefficients as signed measures is proven using a probabilistic approach, which also provides a representation of the solution. Heat kernel estimates and the theory of additive functionals play a crucial role in this approach.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Christian Baer, Lashi Bandara
Summary: We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. We demonstrate the equivalence of various characterisations of elliptic boundary conditions and prove the regularity of solutions up to the boundary. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact, and provide examples treated conveniently by our methods.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Nelson Gutierrez Jimenez, Sergii M. Torba
Summary: The method proposed in this study is based on an analytic approximation of the transmutation operator for approximate solution of initial value and spectral problems for one dimensional Dirac equation. The problem of numerical approximation of solutions is reduced to approximation of the potential matrix by a finite linear combination of matrix valued functions related to generalized formal powers. Convergence rate estimates are proved in terms of smoothness of the potential, allowing for extreme accuracy in computing both lower and higher eigendata.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Haoyang Feng, Xiaokui Yue, Xuechuan Wang
Summary: This paper proposes a uniform linearization-collocation framework for solving initial value and boundary value problems. An optional pre-conditioning procedure is provided to ensure computational accuracy, and rigorous analysis of convergence rate is given. The proposed methods outperform existing solvers and can handle problems that are considered invalid by other methods.
COMPUTER PHYSICS COMMUNICATIONS
(2023)
Article
Mathematics, Applied
Pradip Roul, V. M. K. Prasad Goura
Summary: In this study, a high-order numerical scheme based on B-spline functions is developed for solving a class of nonlinear derivative dependent singular boundary value problems. The method is established through matrix approach and demonstrated to be accurate and robust through consideration of four nonlinear examples. It is more accurate and computationally efficient compared to other numerical schemes using the same B-spline functions.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Review
Computer Science, Artificial Intelligence
Nurettin Dogan, Selami Bayeg, Raziye Mert, Oemer Akin
Summary: In this work, the authors introduce singularly perturbed fuzzy initial value problems (SPFIVPs) and present an algorithm for solving them using the extension principle proposed by Zadeh. They also provide some results on the behavior of the α-cuts of the solutions. To demonstrate the robustness of the algorithm, they fuzzify some examples from the literature and apply the algorithm.
EXPERT SYSTEMS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Jan Nordstrom
Summary: Linearisation is used to analyze nonlinear initial boundary value problems by first converting them into linear problems. This paper attempts to resolve the contradiction between energy conservation and boundedness in the nonlinear and linearised versions of the problem. The authors propose a specific skew-symmetric form and a non-standard linearisation procedure that preserve these properties in the linearised problem and its related dual problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Fredrik Lauren, Jan Nordstrom
Summary: This paper presents a new set of boundary procedures for simulating turbulent boundary layers, which can achieve high accuracy numerical results even on coarse meshes with partial data obtained from a wall model.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Tomas Lundquist, Fredrik Lauren, Jan Nordstrom
Summary: This study combines existing summation-by-parts discretization methods to simplify the numerical framework for solving partial differential equations on complex multi-block/element domains. The interfaces between blocks are treated with inner-product-preserving interpolation operators, resulting in a high-order multi-block operator that encapsulates metric terms and interface treatments. This approach allows for a compact description of the numerical scheme that mimics the continuous counterpart and simplifies stability analysis on multi-block domains.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Jan Nordstrom, Andrew R. Winters
Summary: Boundary conditions and estimates for systems of the nonlinear shallow water equations in two spatial dimensions are derived based on energy and entropy analysis. It is found that the energy method provides more detailed information and is consistent with the entropy analysis. The nonlinear energy analysis reveals the differences between linear and nonlinear analysis and shows that the results from linear analysis may not hold in the nonlinear case. The nonlinear analysis generally requires a different minimal number of boundary conditions compared to the linear analysis, and the magnitude of the flow does not influence the number of required boundary conditions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Jan Nordstrom, Fredrik Lauren
Summary: Energy stable and conservative nonlinear weakly imposed interface conditions for the incompressible Euler equations are derived in the continuous setting. The numerical scheme is proved to be stable and conservative by discretely mimicking the continuous analysis using summation-by-parts operators. The theoretical findings are verified by numerical experiments.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Acoustics
S. Hadi Shamsnia, Sarmad Ghader, S. Abbas Haghshenas, Jan Nordstrom
Summary: This study provides numerical solutions to the two-dimensional linearized shallow water equations using a high-order finite difference scheme in Summation By Parts form. The results show that the wave height error in the vorticity-divergence SWE solutions is orders of magnitude lower compared to the conventional SWE solutions, and all solutions demonstrate stability in numerical tests.
Article
Computer Science, Interdisciplinary Applications
Jan Nordstrom
Summary: We investigate a specific skew-symmetric form of nonlinear hyperbolic problems and find that it leads to bounds on energy and entropy. Taking the compressible Euler equations as an example, we transform them into skew-symmetric form and obtain estimates for energy and entropy. Finally, we demonstrate that the skew-symmetric formulation allows for energy and entropy stable discrete approximations if the scheme is formulated on a summation-by-parts form.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
David A. Kopriva, Gregor J. Gassner, Jan Nordstrom
Summary: This paper presents entropy conserving and dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations, which are relevant in overset mesh methods. The conserving formulation establishes a two-way coupling at the artificial interface boundaries using nonlinear penalty functions that disappear when the solutions coincide. In addition, entropy dissipation and coupling can be optionally introduced through linear penalties in the overlap region.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Correction
Mechanics
Irvy M. A. Gledhill, Hamed Roohani, Karl Forsberg, Peter Eliasson, Beric W. Skews, Jan Nordstrom
THEORETICAL AND COMPUTATIONAL FLUID DYNAMICS
(2023)
Article
Computer Science, Software Engineering
Byron A. Jacobs, Fredrik Lauren, Jan Nordstrom
Summary: In this note, a new numerical method is proposed to approximate the Caputo fractional derivative. By separating the integral and derivative in the fractional derivative components, the error terms from the numerical integration and differentiation are isolated. It is shown that the error is dependent on the order of the fractional derivative and also on the inclusion of the interval end points in the numerical integration. Experimental results demonstrate the validity of the proposed method and how to improve accuracy for smooth functions.
BIT NUMERICAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Alexander Rothkopf, Jan Nordstrom
Summary: Motivated by the connection between classical and quantum physics, this study presents a new discretization prescription for classical initial value problems (IVPs). By formulating the problem as an optimization task, classical trajectories can be obtained without deriving an equation of motion. The demonstrated numerical experiments show the stability, accuracy, and convergence properties of this approach in systems with classical equations of motion.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Jan Glaubitz, Jan Nordstr, Philipp Offner
Summary: Summation-by-parts (SBP) operators are commonly used for developing stable and high-order accurate numerical methods for time-dependent differential equations. This paper presents a theory for SBP operators based on general function spaces and demonstrates that the results for polynomial-based SBP operators can be extended to this general class. The findings show that SBP operators can be applied to a wider range of methods than currently known, using trigonometric, exponential, and radial basis functions as examples.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Computer Science, Interdisciplinary Applications
Jan Glaubitz, Simon-Christian Klein, Jan Nordstroem, Philipp Oeffner
Summary: Summation-by-parts (SBP) operators are used to develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. The existing SBP operators are based on the idea that polynomials can accurately approximate the solution, but this may not always be the best approximation. Function-space SBP (FSBP) operators are developed to address this issue. In this paper, the theory of FSBP operators is extended to multiple dimensions, and it is shown that the concept of SBP operators can be applied to a significantly larger class of methods.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Mojalefa P. Nchupang, Arnaud G. Malan, Fredrik Lauren, Jan Nordstrom
Summary: In this article, a high order accurate method is developed to solve the incompressible boundary layer equations in a provably stable manner. The method utilizes discrete energy estimates and high-order finite difference methods on summation-by-parts form. The newly derived boundary conditions remove singularities associated with the null-space of the nonlinear discrete spatial operator. Numerical experiments confirm the high-order accuracy and consistency with theoretical results.
COMPUTERS & FLUIDS
(2023)
Article
Mathematics, Applied
R. Abgrall, J. Nordstrom, P. Offner, S. Tokareva
Summary: The research finds that under the correct imposition of boundary conditions, a pure continuous Galerkin scheme can be linearly stable, and considering entropy conservation in nonlinear cases can provide an estimation of boundary operators that ensure conservation. The theoretical analysis was validated through numerical simulations.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2023)
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)
