Galerkin methods for the Davey–Stewartson equations

Parallel finite difference schemes for heat equation based upon overlapping domain decomposition

Overview of the Kadomtsev–Petviashvili-hierarchy reduction method for solitons

A novel finite difference discrete scheme for the time fractional diffusion-wave equation

The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.

Two-grid algorithms based on FEM for nonlinear time-fractional wave equations with variable coefficient

Efficient multistep procedures for time-stepping Galerkin methods for nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions are presented and analyzed. The procedures involve using a precoditioned iterative method for approximately solving the differet linear equations arising at each time step in a discrete time Galerkin method. Optimal order convergence rates are obtained for the iterative methods. Work estimates of almost optimal order are obtained.

A phase-based hybridizable discontinuous Galerkin method for the numerical solution of the Helmholtz equation

An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation

Two algorithms used to correct the semi-implicit numerical scheme for time integration of the conservation equations incorporated into the KORSAR/GP computer code two-liquid model are presented. The KORSAR/GP code has been developed jointly by specialists of the Federal State Unitary Enterprise Aleksandrov Research Institute of Technology and Gidropress Experimental Design Office; in 2009, the code was certified at the Federal Service for Environmental, Technological, and Nuclear Supervision (Rostekhnadzor) as applied to numerical safety assessment of VVER-based reactor plants. In the semi-implicit scheme, the convective terms appearing in the phase momentum conservation equations are written in an explicit form. The mass and energy flows of the phases are represented implicitly with respect to phase velocities, and the transferred donor quantities are calculated based on the parameters taken from the previous time layer. Owing to linearization of unsteady and source terms, the linear system of finite difference equations is solved in a noniteration manner. The first of the presented algorithms compensates for numerical imbalances of phase masses and energies resulting from linearization of unsteady terms appearing in the discrete equations and ensures conservativeness of the scheme. The second algorithm introduces correction for the nonphysical redistribution of coolant mass and energy over the computational cells if a change occurs in the phase motion direction within a time step when the scheme of approximating the convective terms becomes “antidonor” with respect to flow. The numerical imbalances and refinements taking into account the correction of donor quantities are computed at each time step according to the proposed correlations and are used for making compensation at the next time step as supplementary sources in the conservation equations. Results from testing the correction algorithms in the KORSAR/GP code are presented. The effectiveness of the imbalance compensation algorithm is confirmed by solving a problem involving the natural circulation loop heating process. The adequacy of the “antidonor” scheme correction algorithm is demonstrated on problems involving steplike initial distribution of the scalar parameters of a stagnant single-phase gaseous or water coolant in a horizontal tube when the flowrate at the tube inlet has different signs at different time steps.

This paper is concerned with the convergence, global superconvergence, local superconvergence, and stability of collocation methods for u′(t) = au(t) + bu([t]). The optimal convergence order and superconvergence order are obtained, and the stability regions for the collocation methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained, and some numerical experiments are given.

In this paper, a stable extrinsic extended finite element method (SXFEM) is proposed to solve the second order elliptic equation with discontinuous coefficients and interfaces. SXFEM is designed by the stable enrichment function and stress intensity factors (SIF)-type enrichment functions. It shows that the proposed SXFEM can get the optimal convergence order. Numerical experiments are presented to verify the feasibility of the new method for this type of problem and the superiority compared with the standard FEM and XFEM.

This article introduces the hybridizable discontinuous Galerkin (HDG) approach to numerically approximate the solution of a linear hyperbolic integro-differential equation. A priori error estimates for semi-discrete and fully discrete schemes are developed. It is shown that the optimal order of convergence is achieved for both scalar and flux variables. To achieve that, an intermediate projection is introduced for the semi-discrete error analysis, and it also shows that this projection achieves convergence of order $h^{k+3/2}$ for $k ≥ 1.$ Next, superconvergence is achieved for the scalar variable using element-by-element post-processing. For the fully discrete error analysis, the central difference scheme and the mid-point rule approximate the derivative and the integral term, respectively. Hence, the second order of convergence is achieved in the temporal direction. Finally, numerical experiments have been performed to validate the theory developed in this article.

A high-order nodal discontinuous Galerkin method for a linearized fractional Cahn–Hilliard equation

In this paper, we apply semi-Lagrangian discontinuous Galerkin (SLDG) methods for linear hyperbolic equations in one space dimension and analyze the error between the numerical and exact solutions under the L 2 L^2 -norm. In all the previous works, the theoretical analysis of the SLDG method would suggest a suboptimal convergence rate due to the error accumulation over time steps. However, numerical experiments demonstrate an optimal convergence rate and, if the terminal time is large, a superconvergence rate. In this paper, we will prove optimal convergence and optimal superconvergence rates. There are three main difficulties: 1. The error analysis on overlapping meshes. Due to the nature of the semi-Lagrangian time discretization, we need to introduce the background Eulerian mesh and the shifted mesh. The two meshes are staggered, and it is not easy to construct local projections and to handle the error accumulation during time evolution. 2. The superconvergence of time-dependent terms under the L 2 L^2 -norm. The error of the numerical and exact solutions can be divided into two parts, the projection error and the time-dependent superconvergence term. The projection strongly depends on the superconvergence rates. Therefore, we need to construct a sequence of projections and gradually improve the superconvergence rates. 3. The stopping criterion of the sequence of projections. The sequence of projections are basically of the same form. We need to show that the projections exist up to some certain order since the superconvergence rate cannot be infinity. Hence, we will seek some “hidden” condition for the existence of the projections. In this paper, we will solve all the three difficulties and construct several local projections to prove the optimal convergence and superconvergence rates. Numerical experiments verify the theoretical findings.

This paper investigates lowest-order weak Galerkin finite element methods for solving linear elasticity problems on rectangular and brick meshes. Specifically, constant vectors are used in element interiors and on element interfaces respectively for approximating displacement. For these constant basis functions, their discrete weak gradients are calculated in the local Raviart–Thomas spaces $$ RT_{[0]}^d $$ ( $$ d=2 $$ or 3), whereas their discrete weak divergences are calculated as elementwise constants. Discrete weak strains are calculated accordingly. Then these quantities are used to develop finite element schemes in both strain-div and grad-div formulations, on both rectangular and brick meshes. A theoretical analysis supported by numerical experiments in both 2-dim and 3-dim reveal that the methods are locking-free and have optimal 1st order convergence in displacement, stress, and dilation (divergence of displacement), when the exact solution has full regularity. The methods can also capture low-regularity solutions very well. Strategies for efficient implementation including Schur complement are presented. Extension to quadrilateral and hexahedral meshes, in both theoretical analysis and numerical experiments, is also examined.