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In this paper, based on the developed nonlinear fourth-order operator and method of order reduction, a novel fourth-order compact difference scheme is constructed for the mixed-type time-fractional Burgers' equation, from which…
We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup…
We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time…
The accuracy of solving partial differential equations (PDEs) on coarse grids is greatly affected by the choice of discretization schemes. In this work, we propose to learn time integration schemes based on neural networks which satisfy…
The numerical analysis of stochastic time fractional evolution equations presents considerable challenges due to the limited regularity of the model caused by the nonlocal operator and the presence of noise. The existing time-stepping…
This paper deals with a new algorithm called modified trigonometric cubic B-spline differential quadrature method for numerical computation of the time dependent partial differential equations. Specially the numerical computation of the…
In this paper, we develop a numerical algorithm for an inverse problem on determining fractional orders of time derivatives simultaneously in a coupled subdiffusion system. Following the theoretical uniqueness, we reformulate the order…
This paper considers one-dimensional heat transfer in a media with temperature-dependent thermal conductivity. To model the transient behavior of the system, we solve numerically the one-dimensional unsteady heat conduction equation with…
Higher-order nonlinear time-evolution equations have widespread applications in science and engineering, such as in solid mechanics, materials science, and fluid mechanics. This paper mainly studies a direct time-parallel algorithm for…
In this paper, we discuss the steady and time-dependent nonlinear convection-diffusion (advection-diffusion) equations with the Dirichlet boundary condition. For the steady nonlinear equation, we use an iteration method to reformulate the…
We propose a high-order spacetime wavelet method for the solution of nonlinear partial differential equations with a user-prescribed accuracy. The technique utilizes wavelet theory with a priori error estimates to discretize the problem in…
We present a second-order ensemble method based on a blended three-step backward differentiation formula (BDF) timestepping scheme to compute an ensemble of Navier-Stokes equations. Compared with the only existing second-order ensemble…
In this work we show that randomized (block) coordinate descent methods can be accelerated by parallelization when applied to the problem of minimizing the sum of a partially separable smooth convex function and a simple separable convex…
In this work, we study the convergence and performance of nonlinear solvers for the Bidomain equations after decoupling the ordinary and partial differential equations of the cardiac system. Firstly, we provide a rigorous proof of the…
We propose novel algorithms combining accelerated gradient flows with linearized projection-free treatments of non-convex constraints and BDF pseudo-temporal discretization for quadratic energy minimization. A general framework is developed…
We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity --- but no…
A fully discrete implicit scheme is proposed for the Swift-Hohenberg model, combining the third-order backward differentiation formula (BDF3) for the time discretization and the second-order finite difference scheme for the space…
In this paper, we first propose an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the $L1$-type formula…
This article focuses on a nonlinear Neumann boundary feedback control formulation for the viscous Burgers' equation and develops a class of finite difference schemes to achieve global stabilization. The proposed procedure, known as the…
An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived from a gradient flow in the negative order Sobolev space $H^{-\alpha}$,…