Related papers: Mathematical base of Difference Operator of Partic…
Physics-informed neural networks (PINNs) and neural operators, two leading scientific machine learning (SciML) paradigms, have emerged as powerful tools for solving partial differential equations (PDEs). Although increasing the training…
By comparing the original equations with the corresponding stationary ones, the moderate deviation principle (MDP) is established for unbounded additive functionals of several different models of distribution dependent SDEs, with…
We study the regularity of the solution to an obstacle problem for a class of integro-differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional…
We construct new, efficient, and accurate high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the…
Discrete mathematics, the study of finite structures, is one of the fastest growing areas in mathematics and optimization. Discrete fractional calculus (DFC) theory that is an important subject of the fractional calculus includes the…
Partial differential equations (PDEs) on surfaces arise in a wide range of applications. The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is a recent embedding method that has been used to solve a…
We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on…
Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential…
The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study…
Dissipative particle dynamics (DPD) is now a well-established method for simulating soft matter systems. However, its applicability was recently questioned because some investigations showed an upper coarse-graining limit that would prevent…
In this paper, a class of finite difference numerical techniques is presented to solve the second-order linear inhomogeneous damped wave equation. The consistency, stability, and convergences of these numerical schemes are discussed. The…
In this paper we develop a dual-support smoothed particle hydrodynamics (DS-SPH) that naturally satisfies the conservation of momentum, angular momentum and energy when the varying smoothing length is utilized. The DS-SPH is based on the…
We develop a computational method based on Dissipative Particle Dynamics (DPD) that introduces solvent hydrodynamic interactions to coarse-grained models of solutes, such as ions, molecules, or polymers. DPD-solvent (DPDS) is a fully…
In this article we introduce a finite difference approximation for integro-differential operators of L\'evy type. We approximate solutions of integro-differential equations, where the second order operator is allowed to degenerate. In the…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
With growing investigations into solving partial differential equations by physics-informed neural networks (PINNs), more accurate and efficient PINNs are required to meet the practical demands of scientific computing. One bottleneck of…
In this paper, we develop a new method for magnetohydrodynamics (MHD) using smoothed particle hydrodynamics (SPH). To describe MHD shocks accurately, the Godunov method is applied to SPH instead of artificial dissipation terms. In the…
Dissipative particle dynamics (DPD) belongs to a class of models and computational algorithms developed to address mesoscale problems in complex fluids and soft matter in general. It is based on the notion of particles that represent…
This paper presents a semi-implicit spectral deferred correction (SDC) method for incompressible Navier-Stokes problems with variable viscosity and time-dependent boundary conditions. The proposed method integrates elements of velocity- and…