482 research outputs found
A Quintic B-Spline Technique for a System of Lane-Emden Equations Arising in Theoretical Physical Applications
In the present study, we introduce a collocation approach utilizing quintic B-spline functions as bases for solving systems of Lane Emden equations which have various applications in theoretical physics and astrophysics. The method derives a solution for the provided system by converting it into a set of algebraic equations with unknown coefficients, which can be easily solved to determine these coefficients. Examining the convergence theory of the proposed method reveals that it yields a fourth-order convergent approximation. It is confirmed that the outcomes are consistent with the theoretical investigation. Tables and graphs illustrate the proficiency and consistency of the proposed method. Findings validate that the newly employed method is more accurate and effective than other approaches found in the literature. All calculations have been performed using Mathematica software
Numerical Solution Of The Heat Equation By Cubic B-Spline Collocation Method
This work proposes a numerical scheme for heat parabolic problem by implementing a collocation method with a cubic B-spline for a uniform mesh. The key idea of this method is to apply forward finite difference and Crank–Nicolson methods for time and space integration, respectively. The stability of the presented scheme is proved through the Von-Neumann technique. It is shown that it is unconditionally stable. The accuracy of the suggested scheme is computed through the L_2 and L_∞-norms. Numerical experiments are also given and show that it is compatible with the exact solutions
A Powerful Robust Cubic Hermite Collocation Method for the Numerical Calculations and Simulations of the Equal Width Wave Equation
In this article, non-linear Equal Width-Wave (EW) equation will be
numerically solved . For this aim, the non-linear term in the equation is
firstly linearized by Rubin-Graves type approach. After that, to reduce the
equation into a solvable discretized linear algebraic equation system which is
the essential part of this study, the Crank-Nicolson type approximation and
cubic Hermite collocation method are respectively applied to obtain the
integration in the temporal and spatial domain directions. To be able to
illustrate the validity and accuracy of the proposed method, six test model
problems that is single solitary wave, the interaction of two solitary waves,
the interaction of three solitary waves, the Maxwellian initial condition,
undular bore and finally soliton collision will be taken into consideration and
solved. Since only the single solitary wave has an analytical solution among
these solitary waves, the error norms Linf and L2 are computed and compared to
a few of the previous works available in the literature. Furthermore, the
widely used three invariants I1, I2 and I3 of the proposed problems during the
simulations are computed and presented. Beside those, the relative changes in
those invariants are presented. Also, a comparison of the error norms Linf and
L2 and these invariants obviously shows that the proposed scheme produces
better and compatible results than most of the previous works using the same
parameters. Finally, von Neumann analysis has shown that the present scheme is
unconditionally stable.Comment: 25 pages, 9 tables, 6 figure
Summation-by-parts operators for general function spaces: The second derivative
Many applications rely on solving time-dependent partial differential
equations (PDEs) that include second derivatives. Summation-by-parts (SBP)
operators are crucial for developing stable, high-order accurate numerical
methodologies for such problems. Conventionally, SBP operators are tailored to
the assumption that polynomials accurately approximate the solution, and SBP
operators should thus be exact for them. However, this assumption falls short
for a range of problems for which other approximation spaces are better suited.
We recently addressed this issue and developed a theory for first-derivative
SBP operators based on general function spaces, coined function-space SBP
(FSBP) operators. In this paper, we extend the innovation of FSBP operators to
accommodate second derivatives. The developed second-derivative FSBP operators
maintain the desired mimetic properties of existing polynomial SBP operators
while allowing for greater flexibility by being applicable to a broader range
of function spaces. We establish the existence of these operators and detail a
straightforward methodology for constructing them. By exploring various
function spaces, including trigonometric, exponential, and radial basis
functions, we illustrate the versatility of our approach. We showcase the
superior performance of these non-polynomial FSBP operators over traditional
polynomial-based operators for a suite of one- and two-dimensional problems,
encompassing a boundary layer problem and the viscous Burgers' equation. The
work presented here opens up possibilities for using second-derivative SBP
operators based on suitable function spaces, paving the way for a wide range of
applications in the future.Comment: 20 pages, 7 figure
Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints
The proximal Galerkin finite element method is a high-order, low iteration
complexity, nonlinear numerical method that preserves the geometric and
algebraic structure of bound constraints in infinite-dimensional function
spaces. This paper introduces the proximal Galerkin method and applies it to
solve free boundary problems, enforce discrete maximum principles, and develop
scalable, mesh-independent algorithms for optimal design. The paper leads to a
derivation of the latent variable proximal point (LVPP) algorithm: an
unconditionally stable alternative to the interior point method. LVPP is an
infinite-dimensional optimization algorithm that may be viewed as having an
adaptive barrier function that is updated with a new informative prior at each
(outer loop) optimization iteration. One of the main benefits of this algorithm
is witnessed when analyzing the classical obstacle problem. Therein, we find
that the original variational inequality can be replaced by a sequence of
semilinear partial differential equations (PDEs) that are readily discretized
and solved with, e.g., high-order finite elements. Throughout this work, we
arrive at several unexpected contributions that may be of independent interest.
These include (1) a semilinear PDE we refer to as the entropic Poisson
equation; (2) an algebraic/geometric connection between high-order
positivity-preserving discretizations and certain infinite-dimensional Lie
groups; and (3) a gradient-based, bound-preserving algorithm for two-field
density-based topology optimization. The complete latent variable proximal
Galerkin methodology combines ideas from nonlinear programming, functional
analysis, tropical algebra, and differential geometry and can potentially lead
to new synergies among these areas as well as within variational and numerical
analysis
A high-performance boundary element method and its applications in engineering
As a semi-numerical and semi-analytical method, owing to the inherent advantage, of boundary-only discretisation, the boundary element method (BEM) has been widely applied to problems with complicated geometries, stress concentration problems, infinite domain problems, and many others. However, domain integrals and non-symmetrical and dense matrix systems are two obstacles for BEM which have hindered the its further development and application. This thesis is aimed at proposing a high-performance BEM to tackle the above two drawbacks and broaden the application scope of BEM. In this thesis, a detailed introduction to the traditional BEM is given and several popular algorithms are introduced or proposed to enhance the performance of BEM. Numerical examples in heat conduction analysis, thermoelastic analysis and thermoelastic fracture problems are performed to assess the efficiency and correction of the algorithms. In addition, necessary theoretical derivations are embraced for establishing novel boundary integral equations (BIEs) for specific engineering problems. The following three parts are the main content of this thesis. (1) The first part (Part II consisting of two chapters) is aimed at heat conduction analysis by BEM. The coefficient matrix of equations formed by BEM in solving problems is fully-populated which occupy large computer memory. To deal with that, the fast multipole method (FMM) is introduced to energize the line integration boundary element method (LIBEM) to performs better in efficiency. In addition, to compute domain integrals with known or unknown integrand functions which are caused by heat sources or heterogeneity, a novel BEM, the adaptive orthogonal interpolation moving least squares (AOIMLS) method enhanced LIBEM, which also inherits the advantage of boundary-only discretisation, is proposed. Unlike LIBEM, which is an accurate and stable method for computing domain integrals, but only works when the mathematical expression of integral function in domain integrals is known, the AOIMLS enhanced LIBEM can compute domain integrals with known or unknown integral functions, which ensures all the nonlinear and nonhomogeneous problems can be solved without domain discretisation. In addition, the AOIMLS can adaptively avoid singular or ill-conditioned moment matrices, thus ensuring the stability of the calculation results. (2) In the second part (Part III consisting of four chapters), the thermoelastic problems and fracture problems are the main objectives. Due to considering thermal loads, domain integrals appear in the BIEs of the thermoelastic problems, and the expression of integrand functions is known or not depending on the temperature distribution given or not, the AOIMLS enhanced LIBEM is introduced to conduct thermoelasticity analysis thereby. Besides, a series of novel unified boundary integral equations based on BEM and DDM are derived for solving fracture problems and thermoelastic fracture problems in finite and infinite domains. Two sets of unified BIEs are derived for fracture problems in finite and infinite domains based on the direct BEM and DDM respectively, which can provide accurate and stable results. Another two sets of BIEs are addressed by employing indirect BEM and DDM, which cannot ensure a stable result, thereby a modified indirect BEM is proposed which performs much more stable. Moreover, a set of novel BIEs based on the direct BEM and DDM for cracked domains under thermal stress is proposed. (3) In the third part (Part IV consisting of one chapter), a high-efficiency combined BEM and discrete element method (DEM) is proposed to compute the inner stress distribution and particle breakage of particle assemblies based on the solution mapping scheme. For the stress field computation of particles with similar geometry, a template particle is used as the representative particle, so that only the related coefficient matrices of one template particle in the local coordinate system are needed to be calculated, while the coefficient matrices of the other particles, can be obtained by mapping between the local and global coordinate systems. Thus, the combined BEM and DEM is much more effective when modelling a large-scale particle system with a small number of distinct possible particle shapes. Furthermore, with the help of the Hoek-Brown criterion, the possible cracks or breakage paths of a particle can be obtained
Effect of the initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water: II: Blowup for nonsmooth conditions.
Abstract Purpose – The purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in shallow water, as a function of the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of three types of initial conditions. Design/methodology/approach – An implicit, first-order accurate in time, finite difference method valid for semipositive relaxation times has been used to solve the equation in a truncated domain for three different initial conditions, a first-order time derivative initially equal to zero and several constant wave speeds. Findings – The numerical experiments show a very rapid transient from the initial conditions to the formation of a leading propagating wave, whose duration depends strongly on the shape, amplitude and width of the initial data as well as on the coefficients of the bidirectional equation. The blowup times for the triangular conditions have been found to be larger than those for the Gaussian ones, and the latter are larger than those for rectangular conditions, thus indicating that the blowup time decreases as the smoothness of the initial conditions decreases. The blowup time has also been found to decrease as the relaxation time, degree of nonlinearity, linear drift coefficient and amplitude of the initial conditions are increased, and as the width of the initial condition is decreased, but it increases as the viscosity coefficient is increased. No blowup has been observed for relaxation times smaller than one-hundredth, viscosity coefficients larger than ten-thousandths, quadratic and cubic nonlinearities, and initial Gaussian, triangular and rectangular conditions of unity amplitude.Funding for open access charge: Universidad de Málaga / CBU
Application of the B-spline Galerkin approach for approximating the time-fractional Burger's equation
This paper presents a numerical scheme based on the Galerkin finite element method and cubic B-spline base function with quadratic weight function to approximate the numerical solution of the time-fractional Burger's equation, where the fractional derivative is considered in the Caputo sense. The proposed method is applied to two examples by using the and error norms. The obtained results are compared with a previous existing method to test the accuracy of the proposed method
GEAR-RT: Towards Exa-Scale Moment Based Radiative Transfer For Cosmological Simulations Using Task-Based Parallelism And Dynamic Sub-Cycling with SWIFT
The development and implementation of GEAR-RT, a radiative transfer solver
using the M1 closure in the open source code SWIFT, is presented, and validated
using standard tests for radiative transfer. GEAR-RT is modeled after RAMSES-RT
(Rosdahl et al. 2013) with some key differences. Firstly, while RAMSES-RT uses
Finite Volume methods and an Adaptive Mesh Refinement (AMR) strategy, GEAR-RT
employs particles as discretization elements and solves the equations using a
Finite Volume Particle Method (FVPM). Secondly, GEAR-RT makes use of the
task-based parallelization strategy of SWIFT, which allows for optimized load
balancing, increased cache efficiency, asynchronous communications, and a
domain decomposition based on work rather than on data. GEAR-RT is able to
perform sub-cycles of radiative transfer steps w.r.t. a single hydrodynamics
step. Radiation requires much smaller time step sizes than hydrodynamics, and
sub-cycling permits calculations which are not strictly necessary to be
skipped. Indeed, in a test case with gravity, hydrodynamics, and radiative
transfer, the sub-cycling is able to reduce the runtime of a simulation by over
90%. Allowing only a part of the involved physics to be sub-cycled is a
contrived matter when task-based parallelism is involved, and is an entirely
novel feature in SWIFT.
Since GEAR-RT uses a FVPM, a detailed introduction into Finite Volume methods
and Finite Volume Particle Methods is presented. In astrophysical literature,
two FVPM methods are written about: Hopkins (2015) have implemented one in
their GIZMO code, while the one mentioned in Ivanova et al. (2013) isn't used
to date. In this work, I test an implementation of the Ivanova et al. (2013)
version, and conclude that in its current form, it is not suitable for use with
particles which are co-moving with the fluid, which in turn is an essential
feature for cosmological simulations.Comment: PhD Thesi
Vandermonde Neural Operators
Fourier Neural Operators (FNOs) have emerged as very popular machine learning
architectures for learning operators, particularly those arising in PDEs.
However, as FNOs rely on the fast Fourier transform for computational
efficiency, the architecture can be limited to input data on equispaced
Cartesian grids. Here, we generalize FNOs to handle input data on
non-equispaced point distributions. Our proposed model, termed as Vandermonde
Neural Operator (VNO), utilizes Vandermonde-structured matrices to efficiently
compute forward and inverse Fourier transforms, even on arbitrarily distributed
points. We present numerical experiments to demonstrate that VNOs can be
significantly faster than FNOs, while retaining comparable accuracy, and
improve upon accuracy of comparable non-equispaced methods such as the Geo-FNO.Comment: 21 pages, 10 figure
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