相关论文: Explicit and implicit TVD schemes for conservation…

The positive definiteness of discrete time-fractional derivatives is fundamental to the numerical stability (in the energy sense) for time-fractional phase-field models. A novel technique is proposed to estimate the minimum eigenvalue of…

We develop a second-order well-balanced central-upwind scheme for the compressible Euler equations with gravitational source term. Here, we advocate a new paradigm based on a purely conservative reformulation of the equations using global…

In this article, we present a new second order finite difference discrete scheme for fractal mobile/immobile transport model based on equivalent transformative Caputo formulation. The new transformative formulation takes the singular kernel…

Fractional differential equations model processes with memory effects, providing a realistic perspective on complex systems. We examine time-delayed differential equations, discussing first-order and fractional Caputo time-delayed…

In this note, we prove or re-prove several important results regarding one dimensional time fractional ODEs following our previous work \cite{fllx17}. Here we use the definition of Caputo derivative proposed in \cite{liliu17frac1,liliu2017}…

A novel H3N3-2$_\sigma$ interpolation approximation for the Caputo fractional derivative of order $\alpha\in(1,2)$ is derived in this paper, which improves the popular L2C formula with (3-$\alpha$)-order accuracy. By an interpolation…

In the case of scalar conservation laws $$ u_{t} + f(u)_{x}~=~0,\qquad t\geq 0, x\in\mathbb{R}, $$ with uniformly strictly convex flux $f$, quantitative compactness estimates - in terms of Kolmogorov entropy in ${\bf L}^{1}_{loc}$ - were…

We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is…

This paper proposes a novel preconditioned implicit-explicit algorithm enhanced with the extrapolation technique for non-convex optimization problems. The algorithm employs a third-order Adams-Bashforth scheme for the nonlinear and explicit…

An "exact" method for scalar one-dimensional hyperbolic conservation laws is presented. The approach is based on the evolution of shock particles, separated by local similarity solutions. The numerical solution is defined everywhere, and is…

The aim of this work is to design implicit and semi-implicit high-order well-balanced finite-volume numerical methods for 1D systems of balance laws. The strategy introduced by two of the authors in a previous paper for explicit schemes…

In this work, we consider a time-fractional Allen-Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order $\alpha\in(0,1)$. First, the well-posedness and (limited) smoothing…

A numerical analysis of the effect of artificial viscosity is undertaken in order to understand the effect of numerical diffusion on numerical boundary feedback control. The analysis is undertaken on the linear hyperbolic systems…

Safety exploration can be regarded as a constrained Markov decision problem where the expected long-term cost is constrained. Previous off-policy algorithms convert the constrained optimization problem into the corresponding unconstrained…

The existing discrete variational derivative method is only second-order accurate and fully implicit. In this paper, we propose a framework to construct an arbitrary high-order implicit (original) energy stable scheme and a second-order…

This paper focuses on the numerical approximation of the solutions of non-local conservation laws in one space dimension. These equations are motivated by two distinct applications, namely a traffic flow model in which the mean velocity…

The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian…

This paper proposes a new class of arbitrarily high-order conservative numerical schemes for the generalized Korteweg-de Vries (KdV) equation. This approach is based on the scalar auxiliary variable (SAV) method. The equation is…

A new discrete energy dissipation law of the variable-step fractional BDF2 (second-order backward differentiation formula) scheme is established for time-fractional Cahn-Hilliard model with the Caputo's fractional derivative of order…

This short study consists of two parts, firstly we obtain some inequalities on Caputo Fractional derivatives using the elementary inequalities. Secondly we establish several new inequalities including Caputo fractional derivatives for…