Three-level Difference Scheme Research Articles

Unsteady hydrodynamics of a surface piercing and fully submerged body when entering a lock

Ship hydrodynamics in a constant waterway have been well studied. With a coordinate system fixed on the moving ship, the boundary value problem (BVP) is usually treated as a steady one. However, the hydrodynamics of a ship moving in confined waterways with abrupt changes in width or depth are complex due to their unsteady nature. The studies on such unsteady problems are insufficient. In the engineering practice, the hydrodynamic unsteadiness can be fully reflected by a scenario when a ship enters a lock. Prior studies have predominantly focused on predicting the ships' hydrodynamic forces without considering the unsteady terms on the free-surface boundary conditions. Obviously, such steady or quasi-steady methods overlooked the crucial unsteady phenomena on free water surface. To address this gap, the present study introduces a novel three-level difference scheme to discretize the free surface condition, preserving unsteady terms while maintaining temporal continuity of cells on the free surface. With the implementation of such fully unsteady BVP, we observed some interesting unsteady free surface motions, which were not well documented in the existing literature. To verify our new observations, as well as to validate the numerical method proposed in this study, two physical model tests were designed and conducted in a towing tank: a submerged ellipsoid enters a deep lock at relatively high speeds, and a box enters a shallow and narrow lock at very low speeds. The discussion is highlighted on the unsteady waves in front of the moving bodies, as well as the unsteady resistance induced by such waves.

A three-level implicit difference scheme for solving the inviscid Burgers' equation with time delay

A three-level implicit difference scheme is developed for a inviscid Burgers' equation with time delay. It is proved that the proposed scheme is uniquely solvable, stable, and unconditionally convergent with order in the maximal norm based on an assumption of monotone initial data and smooth solutions, where τ is the temporal step size and h is the spatial step size. Finally, numerical experiments are applied to illustrate the effectiveness of theoretical results.

Spectral Stability Conditions for an Explicit Three-Level Finite-Difference Scheme for a Multidimensional Transport Equation with Perturbations

We study difference schemes associated with a simplified linearized multidimensionalhyperbolic quasi-gasdynamic system of differential equations. It is shown that an explicit two-levelvector difference scheme with flux relaxation for a second-order hyperbolic equation with variablecoefficients that is a perturbation of the transport equation with a parameter multiplying thehighest derivatives can be reduced to an explicit three-level difference scheme. In the case ofconstant coefficients, the spectral condition for the time-uniform stability of this explicitthree-level difference scheme is analyzed, and both sufficient and necessary conditions for thiscondition to hold are derived, in particular, in the form of Courant type conditions on the ratio oftemporal and spatial steps.

A finite difference scheme for solving the undamped Timoshenko beam equations with both ends free

In this paper, a boundary feedback system of a class of non-uniform undamped Timoshenko beam with both ends free is considered. A linearized three-level difference scheme for the Timoshenko beam equations is derived by the method of reduction of order on uniform meshes. The unique solvability, unconditional stability and convergence of the difference scheme are proved by the discrete energy method. The convergence order in maximum norm is of order two in both space and time. The validity of this theoretical analysis is verified experimentally.

Cubic Spline Method for 1D Wave Equation in Polar Coordinates

Using nonpolynomial cubic spline approximation in space and finite difference in time direction, we discuss three-level implicit difference scheme of O(k2+h4) for the numerical solution of 1D wave equations in polar coordinates, where k>0 and h>0 are grid sizes in time and space coordinates, respectively. The proposed method is applicable to problems with singularity. Stability theory of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

New conservative difference schemes for a coupled nonlinear Schrödinger system

In this paper, two conservative difference schemes for solving a coupled nonlinear Schrödinger (CNLS) system are numerically analyzed. Firstly, a nonlinear implicit two-level finite difference scheme for CNLS system is studied, then a linear three-level difference scheme for CNLS system is presented. An induction argument and the discrete energy method are used to prove the second-order convergence and unconditional stability of the linear scheme. Numerical examples show the efficiency of the new scheme.

An unconditionally stable spline difference scheme of [formula omitted] for solving the second-order 1D linear hyperbolic equation

In this paper, the second-order linear hyperbolic equation is solved by using a new three-level difference scheme based on quartic spline interpolation in space direction and finite difference discretization in time direction. Stability analysis of the scheme is carried out. The proposed scheme is second-order accurate in time direction and fourth-order accurate in space direction. Finally, numerical examples are tested and results are compared with other published numerical solutions.

Error estimates for a projection-difference method for a linear differential-operator equation

We study a projection-difference method for approximately solving the Cauchy problem u′(t) + A(t)u(t) + K(t)u(t) = h(t), u(0) = 0 for a linear differential-operator equation in a Hilbert space, where A(t) is a self-adjoint operator and K(t) is an operator subordinate to A(t). Time discretization is based on a three-level difference scheme, and space discretization is carried out by the Galerkin method. Under certain smoothness conditions on the function h(t), we obtain estimates for the convergence rate of the approximate solutions to the exact solution.

A difference scheme for solving the Timoshenko beam equations with tip body

In this article, a Timoshenko beam with tip body and boundary damping is considered. A linearized three-level difference scheme of the Timoshenko beam equations on uniform meshes is derived by the method of reduction of order. The unique solvability, unconditional stability and convergence of the difference scheme are proved. The convergence order in maximum norm is of order two in both space and time. A numerical example is presented to demonstrate the theoretical results.

The stability and convergence of an explicit difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions

This article is concerned with the numerical solution to the time-dependent Schrödinger equation on an infinite domain. Two exact artificial boundary conditions are introduced to reduce the original problem into an initial boundary value problem with a finite computational domain. The artificial boundary conditions involve the 1/2 order fractional derivative in t. Then, a fully discrete explicit three-level difference scheme is derived. The truncation errors are analyzed in detail. The stability and convergence with the convergence order of O(h3/2+τh−1/2) are proved under the condition τ/h2<1/2 by the energy method. A numerical example is given to demonstrate the accuracy and efficiency of the proposed method. Two open problems are brought forward at the end of the article.

Difference Schemes for Nonstationary Vector Problems

So far, twoand three-level difference schemes have been constructed and investigated for basic nonstationary problems of mathematical physics [1, 2]. We mainly speak of initial-boundary value problems for a second-order parabolic equation (the heat equation). In this case, the elliptic operator with respect to the space variables may be self-adjoint or nonself-adjoint, as in the convection–diffusion equation [3]. Similar results can be obtained for second-order hyperbolic equations (the oscillation equation). In the nonself-adjoint case, unconditionally stable schemes are most often constructed under the condition that the nonself-adjoint part of the operator is subordinate to the self-adjoint part [2]. These results can be generalized in the theory of operator-difference schemes, where operator statements in various spaces for the space derivatives are used in the initial-boundary value problems of mathematical physics [1, 2]. This permits one to consider various classes of nonstationary problems of mathematical physics from a common viewpoint by neglecting, in particular, the specific nature of differential equations and boundary conditions. Results on stability with respect to the initial data and the right-hand side and stability under perturbations of the problem operator (the coefficients of the original equation) were obtained in [4] for twoand three-level operatordifference schemes. In the approximate solution of initial-boundary value problems for multidimensional partial differential equations, special attention is paid to the construction of additive schemes, that is, decomposition schemes [1, 5, 6]. The passage to a chain of simpler problems permits one to construct efficient difference schemes (decomposition with respect to the space variables). In a number of cases, it is useful to extract subproblems of different nature (decomposition with respect to physical processes). Regionally additive schemes (domain decomposition schemes) designed for the construction of numerical algorithms for parallel computers [6] have been actively discussed in recent time. In the case of a multicomponent decomposition (into three or more operators), unconditionally stable additive schemes are usually constructed with the use of the notion of total approximation on the basis of the passage to a chain of separate initial value problems for each operator term. In a number of cases, additive schemes of multicomponent decomposition are constructed without using the total approximation (so-called regularized additive schemes [6]). We also single out a new class of additive schemes for vector problems. Such schemes can be used in the construction of effective numerical algorithms for the approximate solution of systems of equations of mathematical physics, that is, vector equations. Typical is the situation in which separate components of the vector of unknowns are coupled with each other, and it is difficult to pose a good problem for the components on a new time level. Note some examples of the construction of additive schemes for vector problems. In the application of additive difference schemes to systems of parabolic and hyperbolic equations in [1], the passage to simpler problems is performed on the basis of the regularization principle. The regularization principle was suggested in [7], where its capabilities were illustrated by a number of meaningful examples. The most impressive results [1, 2] were obtained in the construction of unconditionally stable difference schemes for nonstationary problems of mathematical physics, factorized schemes for multidimensional problems, and iterative methods for grid problems [8]. One improves the properties of a difference scheme by perturbing the operators of the

On the numerical solution of quasilinear wave equation with strong dissipative term

The numerical solution for a type of quasilinear wave equation is studied. The three-level difference scheme for quasi-linear waver equation with strong dissipative term is constructed and the convergence is proved. The error of the difference solution is estimated. The theoretical results are controlled on a numerical example.

An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation

An implicit three-level difference scheme of O( k 2 + h 2) is discussed for the numerical solution of the linear hyperbolic equation u tt + 2 αu t + β 2 u = u xx + f( x, t), α > β ≥ 0, in the region Ω = {(x,t) ∥ 0 < x < 1, t > 0} subject to appropriate initial and Dirichlet boundary conditions, where α and β are real numbers. We have used nine grid points with a single computational cell. The proposed scheme is unconditionally stable. The resulting system of algebraic equations is solved by using a tridiagonal solver. Numerical results demonstrate the required accuracy of the proposed scheme.

A second-order linearized difference scheme on nonuniform meshes for nonlinear parabolic systems with Dirichlet boundary value conditions

A linearized three-level difference scheme on nonuniform meshes is derived by the method of the reduction of order for the Dirichlet boundary value problem of the nonlinear parabolic systems. It is proved that the difference scheme is uniquely solvable and second order convergent in L∞-norm. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 638–652, 2003

Three-Level Difference Schemes on Non-Uniform in Time Grids

Abstract In this work, a stability of three-level operator-difference schemes on nonuniform in time grids in Hilbert spaces is studied. A priori estimates of a long time stability (for t → ∞) in the sense of the initial data and the right-hand side are obtained in different energy norms without demanding the quasiuniformity of the grid. New difference schemes of the second order of local approximation on nonuniform grids both in time and space on standard stencils for parabolic and wave equations are adduced.

Stability and Regularization of Three-Level Difference Schemes with Unbounded Operator Coefficients in Banach Spaces

The problem of stability of difference schemes for second-order evolution problems is considered. Difference schemes are treated as abstract Cauchy problems for difference equations with operator coefficients in a Banach or Hilbert space. To construct stable difference schemes the regularization principle is employed, i.e., one starts from any simple scheme (possibly unstable) and derives absolutely stable schemes by perturbing the operator coefficients. The main result of this paper is the following: for the first time sufficient conditions are pointed out under which an unstable three-level difference scheme with unbounded operator coefficients in a Banach space can be regularized to a stable scheme. The principal stability condition is the strong P-positivity of the unbounded operator coefficients.

A Conservative Spectral Method for Several Two-Dimensional Nonlinear Wave Equations

A conservative spectral method is proposed to solve several two-dimensional nonlinear wave equations. The conventional fast Fourier transform is used to approximate the spatial derivatives and a three-level difference scheme with a free parameter θ is to advance the solution in time. Our time discretization is semi-implicit in the sense that the linear terms are treated implicitly while the nonlinear terms are evaluated only by previous time levels, and thus treated explicitly. However, the cost of the algorithm is no greater than that of a fully explicit method because the linear boundary value problem that must be solved at each time step is almost trivial in a spectral spatial discretization. A linear stability analysis shows that the method leads to a less restrictive stability condition than the corresponding explicit one. The method is conservative and the ratio of the numerical dispersion to the physical dispersion is of the order O(Δ t2). Applications of our method to the Kadomtsev–Petviashvili and the Zakharov–Kuznetsov equations exhibit excellent results.

On Convergence and Stability of the Explicit Difference Method for Solution of Nonlinear Schrödinger Equations

The first boundary value problem for a nonlinear Schrodinger equation is investigated. The conditional convergence and stability on the initial data of the explicit three-level difference scheme of DuFort--Frankel type in C and W_2^1$ norms are proved. Grid analogues of energy conservation laws and grid multiplicative inequalities are used.

A new high-order accuracy explict difference scheme for solving three-dimensional parabolic equations

In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r=Δt/Δx2=Δt/Δy2=Δt/Δx2<-1/4, and the truncation error is O(Δt2+Δx4).

Accuracy and stability analysis of numerical schemes for the shallow water model

The accuracy and stability properties of several two-level and three-level difference schemes for solving the shallow water model are analyzed by the linearized Fourier Method. The effects of explicit or implicit treatments of the gravity, Coriolis, convective and friction terms on accuracy and stability are examined. The use of Miller's properties on von Neumann polynomial plays a crucial role to resolve the tedious mathematical procedures in the Fourier analysis. As a best compromise between efficiency and stability, we recommend the semi-implicit schemes, where the surface elevation and friction terms are treated implicitly while the convective and Coriolis terms are treated explicitly. © 1996 John Wiley & Sons, Inc.