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We investigate theoretically and numerically the use of the Least-Squares Finite-element method (LSFEM) to approach data-assimilation problems for the steady-state, incompressible Navier-Stokes equations. Our LSFEM discretization is based…
The Lane-Emden type equations are employed in the modelling of several phenomena in the areas of mathematical physics and astrophysics . In this paper a new numerical method is applied to investigate some well-known classes of Lane-Emden…
Error bounds for fully discrete schemes for the evolutionary incompressible Navier--Stokes equations are derived in this paper. For the time integration we apply BDF-$q$ methods, $q\le 5$, for which error bounds for $q\ge 3$ cannot be found…
Efficient algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the curse of dimensionality. We extend the forward-backward stochastic neural networks…
Loosely speaking, the Navier-Stokes-$\alpha$ model and the Navier-Stokes equations differ by a spatial filtration parametrized by a scale denoted $\alpha$. Starting from a strong two-dimensional solution to the Navier-Stokes-$\alpha$ model…
In this work, we propose an adaptive radial basis function (RBF) approach for the efficient solution of multidimensional spatiotemporal integrodifferential equations. Our approach can automatically adjust the shape of RBFs and provide an…
We describe and test numerically an adaptive meshless generalized finite difference method based on radial basis functions that competes well with the finite element method on standard benchmark problems with reentrant corners of the…
In this paper, a multiple-distribution-function finite-difference lattice Boltzmann method (MDF-FDLBM) is proposed for the convection-diffusion system based incompressible Navier-Stokes equations (NSEs). By Chapman Enskog analysis, the…
This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial…
Introduction: the Navier-Stokes equations are essential in fluid dynamics, describing the motion of fluids like liquids and gases. Solving these equations, especially in complex flows and high-Reynolds-number regimes, is a significant…
Compared with conventional numerical approaches to solving partial differential equations (PDEs), physics-informed neural networks (PINN) have manifested the capability to save development effort and computational cost, especially in…
In this paper, we address the problem of predicting complex, nonlinear spatiotemporal dynamics when available data is recorded at irregularly-spaced sparse spatial locations. Most of the existing deep learning models for modeling…
We consider error estimates for the fully discretized instationary Navier-Stokes problem. For the spatial approximation we use conforming inf-sup stable finite element methods in conjunction with grad-div and local projection stabilization…
When numerically computing high Reynolds number cavity flow, it is known that by formulating the Navier-Stokes equations using the stream function and vorticity as unknown functions, it is possible to reproduce finer flow structures.…
Accurate interpolation of functions and derivatives is crucial in solving partial differential equations (PDEs). The Radial Basis Function (RBF) method has become an extremely popular and robust approach for interpolation on scattered data.…
This article aims to develop a direct numerical approach to solve the space-fractional partial differential equations (PDEs) based on a new differential quadrature (DQ) technique. The fractional derivatives are approximated by the weighted…
Radial Basis Function Networks (RBFNs) are used primarily to solve curve-fitting problems and for non-linear system modeling. Several algorithms are known for the approximation of a non-linear curve from a sparse data set by means of RBFNs.…
We present a class of hybrid FD-FV (finite difference and finite volume) methods for solving general hyperbolic conservation laws written in first-order form. The presentation focuses on one- and two-dimensional Cartesian grids; however,…
In this paper we obtain approximated numerical solutions for the 2D Helmholtz equation using a radial basis function-generated finite difference scheme (RBF-FD), where weights are calculated by applying an oscillatory radial basis function…
In this paper we derive local estimates of solutions of the Perturbed Stokes system. This system arises as a reduction of the Stokes system near a curved part of the boundary of the domain if one applies a diffeomorphism flatting the…