High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation

In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.

Quartic Non-Polynomial Spline for Solving the Third-Order Dispersive Partial Differential Equation

In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion processes, hydrodynamics, aerodynamics, etc. These problems have various important applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary differential equation. Truncation errors are given. The unconditional stability of the method is analysed by the Von-Neumann stability analysis. The developed method is tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, a graphical comparison between analytical and approximate solutions is also shown for the illustrated example.

Comparing Solutions to the Nonlinear Dissipative Wave Equation

In previous decades, many of the practical problems arising in scientific fields such as physics, engineering, and mathematics have been related to nonlinear fractional partial differential equations. One of these nonlinear partial differential equations, the dissipative wave equation, has been found to have a plethora of useful applications in different fields. A special class of solutions has been studied for the dissipative wave equation including exact solutions and approximate solutions. The aim of this article is to compare the non-polynomial spline method and the cubic B-spline method with the solution of a nonlinear dissipative wave equation. We will conduct a comparison of the stability of the two methods using the Von Neumann stability analysis. In addition, a numerical example will be presented to illustrate the accuracy of these methods.

An Automatic Analysis Approach Toward Indistinguishability of Sampling on the LWE Problem

Learning With Errors (LWE) is one of the Non-Polynomial (NP)-hard problems applied in cryptographic primitives against quantum attacks.However,the security and efficiency of schemes based on LWE are closely affected by the error sampling algorithms.The existing pseudo-random sampling methods potentially have security leaks that can fundamentally influence the security levels of previous cryptographic primitives.Given that these primitives are proved semantically secure,directly deducing the influences caused by leaks of sampling algorithms may be difficult.Thus,we attempt to use the attack model based on automatic learning system to identify and evaluate the practical security level of a cryptographic primitive that is semantically proved secure in indistinguishable security models.In this paper,we first analyzed the existing major sampling algorithms in terms of their security and efficiency.Then,concentrating on the Indistinguishability under Chosen-Plaintext Attack (IND-CPA) security model,we realized the new attack model based on the automatic learning system.The experimental data demonstrates that the sampling algorithms perform a key role in LWE-based schemes with significant disturbance of the attack advantages,which may potentially compromise security considerably.Moreover,our attack model is achievable with acceptable time and memory costs.

TheUltraWeakVariational FormulationUsing Bessel Basis Functions

We investigate the ultra weak variational formulation(UWVF)of the 2-D Helmholtz equation using a new choice of basis functions.Traditionally the UWVF basis functions are chosen to be plane waves.Here,we instead use first kind Bessel functions.We compare the performance of the two bases.Moreover,we show that it is possible to use coupled plane wave and Bessel bases in the same mesh.As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.