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This work presents a numerical analysis of a master equation modeling the interaction of a system with a noisy environment in the particular context of open quantum systems. It is shown that our transformed master equation has a reduced computational cost in comparison to a Wigner-Fokker-Planck model of the same system for the general case of any potential. Specifics of a NIPG-DG numerical scheme adequate for the convection-diffusion system obtained are then presented. This will let us solve computationally the transformed system of interest modeling our open quantum system. A benchmark problem, the case of a harmonic potential, is then presented, for which the numerical results are compared against the analytical steady-state solution of this problem.

We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schr\"odinger equation (NLSE) with semi-smooth nonlinearity $ f(\rho) = \rho^\sigma$, where $\rho=|\psi|^2$ is the density with $\psi$ the wave function and $\sigma>0$ is the exponent of the semi-smooth nonlinearity. Under the assumption of $ H^2 $-solution of the NLSE, we prove error bounds at $ O(\tau^{\frac{1}{2}+\sigma} + h^{1+2\sigma}) $ and $ O(\tau + h^{2}) $ in $ L^2 $-norm for $0<\sigma\leq\frac{1}{2}$ and $\sigma\geq\frac{1}{2}$, respectively, and an error bound at $ O(\tau^\frac{1}{2} + h) $ in $ H^1 $-norm for $\sigma\geq \frac{1}{2}$, where $h$ and $\tau$ are the mesh size and time step size, respectively. In addition, when $\frac{1}{2}<\sigma<1$ and under the assumption of $ H^3 $-solution of the NLSE, we show an error bound at $ O(\tau^{\sigma} + h^{2\sigma}) $ in $ H^1 $-norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional $ L^2 $-stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of $ 0 < \sigma \leq \frac{1}{2}$, and to establish an $ l^\infty $-conditional $ H^1 $-stability to obtain the $ l^\infty $-bound of the numerical solution by using the mathematical induction and the error estimates for the case of $ \sigma \ge \frac{1}{2}$; and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.

We review how to simulate continuous determinantal point processes (DPPs) and improve the current simulation algorithms in several important special cases as well as detail how certain types of conditional simulation can be carried out. Importantly we show how to speed up the simulation of the widely used Fourier based projection DPPs, which arise as approximations of more general DPPs. The algorithms are implemented and published as open source software.

This study focuses on addressing the challenges of solving analytically intractable differential equations that arise in scientific and engineering fields such as Hamilton-Jacobi-Bellman. Traditional numerical methods and neural network approaches for solving such equations often require independent simulation or retraining when the underlying parameters change. To overcome this, this study employs a physics-informed DeepONet (PI-DeepONet) to approximate the solution operator of a nonlinear parabolic equation. PI-DeepONet integrates known physics into a deep neural network, which learns the solution of the PDE.

Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models.

In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman--Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.

This work is concerned with kinetic equations with velocity of constant magnitude. We propose a quadrature method of moments based on the Poisson kernel, called Poisson-EQMOM. The derived moment closure systems are well defined for all physically relevant moments and the resultant approximations of the distribution function converge as the number of moments goes to infinity. The convergence makes our method stand out from most existing moment methods. Moreover, we devise a delicate moment inversion algorithm. As an application, the Vicsek model is studied for overdamped active particles. Then the Poisson-EQMOM is validated with a series of numerical tests including spatially homogeneous, one-dimensional and two-dimensional problems.

A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not constitute dynamical systems. In several physical systems, we encounter integro-differential equations with memory terms where the time derivative of a state variable at a given time depends on all past states of the system. Secondly, there are equations whose solutions do not have well-defined Taylor series expansion. The Maxey-Riley-Gatignol equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, displays both challenges. We use it as a test bed to address the questions we raise, but our method may be applied to all equations of this class. We show that the Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system which is constructed by introducing a new dynamical co-evolving state variable that encodes memory of past states. We develop a Runge-Kutta algorithm for the resultant Markovian system. The form of the kernels involved in deriving the Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$. Our approach naturally inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition.

Simulation of wave propagation in poroelastic half-spaces presents a common challenge in fields like geomechanics and biomechanics, requiring Absorbing Boundary Conditions (ABCs) at the semi-infinite space boundaries. Perfectly Matched Layers (PML) are a popular choice due to their excellent wave absorption properties. However, PML implementation can lead to problems with unknown stresses or strains, time convolutions, or PDE systems with Auxiliary Differential Equations (ADEs), which increases computational complexity and resource consumption. This article presents two new PML formulations for arbitrary poroelastic domains. The first formulation is a fully-mixed form that employs time-history variables instead of ADEs, reducing the number of unknowns and mathematical operations. The second formulation is a hybrid form that restricts the fully-mixed formulation to the PML domain, resulting in smaller matrices for the solver while preserving governing equations in the interior domain. The fully-mixed formulation introduces three scalar variables over the whole domain, whereas the hybrid form confines them to the PML domain. The proposed formulations were tested in three numerical experiments in geophysics using realistic parameters for soft sites with free surfaces. The results were compared with numerical solutions from extended domains and simpler ABCs, such as paraxial approximation, demonstrating the accuracy, efficiency, and precision of the proposed methods. The article also discusses the applicability of these methods to complex media and their extension to the Multiaxial PML formulation. The codes for the simulations are available for download from \url{//github.com/hmella/POROUS-HYBRID-PML}.

This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in $L^1(\mathbb{R})$ of the numerical solution. Our results partially answer positively to the open problem emerged in [J. Cui and J. Hong, J. Differential Equations (2020)] on computing the density of the exact solution numerically.