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Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs), also known as ODE filters, have been established as efficient methods for quantifying numerical uncertainty in the solution of ODEs. In practical…
We present a pseudo-spectral method for solving the three-dimensional Boussinesq equations in unbounded cylindrical domains, specifically tailored for rotating, stably stratified flows subject to strong azimuthal shear. To effectively…
An efficient algorithm for solving Poisson's equation in two and three spatial dimensions is discussed. The algorithm, which is described in detail, is based on the integral form of Poisson's equation and utilizes spherical coordinates and…
We present a dimensionally split method for solving hyperbolic conservation laws on Cartesian cut cell meshes. The approach combines local geometric and wave speed information to determine a novel stabilised cut cell flux, and we provide a…
In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD) system, by adopting an intrinsically hydrodynamic approach. The present work continues the analysis initiated in [6] where the one dimensional case was studied.…
We propose a new algorithm to the problem of polygonal curve approximation based on a multiresolution approach. This algorithm is suboptimal but still maintains some optimality between successive levels of resolution using dynamic…
Solutions of certain partial differential equations (PDEs) are often represented by the steepest descent curves of corresponding functionals. Minimizing movement scheme was developed in order to study such curves in metric spaces.…
Semi-implicit methods are powerful and efficient tools for the three-dimensional modeling of coastal and oceanic processes. A semi-implicit finite difference method for 3D hydrostatic primitive equations is presented in this paper. The…
In this work we study the dynamic behaviour of compound shells of revolution partially filled with an ideal incompressible fluid based on boundary-value problems. New analytical mathematical model with corresponding discrete scheme for the…
The theory of quadrature domains for harmonic functions and the Hele-Shaw problem of the fluid dynamics are related subjects of the complex variables and mathematical physics. We present results generalizing the above subjects for elliptic…
Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and…
A new methodology is developed to integrate numerically the equations of motion for classical many-body systems in molecular dynamics simulations. Its distinguishable feature is the possibility to preserve, independently on the size of the…
Recently, the deep learning method has been used for solving forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). It has good accuracy and performance for high-dimensional…
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…
A new gridding technique for the solution of partial differential equations in cubical geometry is presented. The method is based on volume penalization, allowing for the imposition of a cubical geometry inside of its circumscribing sphere.…
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…
For static reductions of isotropic and anisotropic Magnetohydrodynamics plasma equilibrium models, a complete classification of admitted point symmetries and conservation laws up to first order is presented. It is shown that the symmetry…
Quadratization refers to a transformation of an arbitrary system of polynomial ordinary differential equations to a system with at most quadratic right-hand side. Such a transformation unveils new variables and model structures that…
This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a new developed…