scholarly journals Analysis of splitting schemes for 2D and 3D Schrödinger problems

2012 ◽  
Vol 53 ◽  
Author(s):  
Aleksas Mirinavičius
Keyword(s):  
Numerical Experiments ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Splitting Scheme ◽  
Splitting Schemes ◽  
Unconditionally Stable ◽  
L2 Norm ◽  
The Stability ◽  
2D And 3D ◽  
Discrete Conservation Laws

New splitting finite difference schemes for 2D and 3D linear Schrödinger problems are investigated. The stability and convergence analysis is done in the discrete L2 norm. It is proved that the 2D scheme is unconditionally stable and conservative in the case of zero boundary condition. The splitting scheme is generalized for 3D problems. It is proved that in this case the scheme is only ρ-stable and consequently discrete conservation laws are no longer valid. Results of numerical experiments are presented.

The Finite Difference Scheme for 3d Mathematical Modeling of a Wood Drying Process

2001 ◽  
Vol 1 (2) ◽  
pp. 125-137 ◽  
Author(s):  
Raimondas Čiegis ◽  
Vadimas Starikovičius
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Drying Process ◽  
The Stability ◽  
Robin Boundary

AbstractThis work discusses issues on the design and analysis of finite difference schemes for 3D modeling the process of moisture motion in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for Robin boundary conditions. The influence of boundary conditions is investigated, and results of numerical experiments are presented.

scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

scholarly journals Calibration of the Volatility in Option Pricing Using the Total Variation Regularization

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yu-Hua Zeng ◽  
Shou-Lei Wang ◽  
Yu-Fei Yang
Keyword(s):  
Total Variation ◽  
Numerical Experiments ◽  
Implied Volatility ◽  
Lagrange Equation ◽  
Stability And Convergence ◽  
Necessary Condition ◽  
Total Variation Regularization ◽  
Future Risk ◽  
The Stability ◽  
Regularization Strategy

In market transactions, volatility, which is a very important risk measurement in financial economics, has significantly intimate connection with the future risk of the underlying assets. Identifying the implied volatility is a typical PDE inverse problem. In this paper, based on the total variation regularization strategy, a bivariate total variation regularization model is proposed to estimate the implied volatility. We not only prove the existence of the solution, but also provide the necessary condition of the optimal control problem—Euler-Lagrange equation. The stability and convergence analyses for the proposed approach are also given. Finally, numerical experiments have been carried out to show the effectiveness of the method.

The Maximum Principle and Some of Its Applications

2002 ◽  
Vol 2 (1) ◽  
pp. 50-91 ◽  
Author(s):  
Piotr Matus
Keyword(s):  
Maximum Principle ◽  
Order Approximation ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
The Maximum Principle ◽  
Stability Of Difference Schemes ◽  
Mixed Derivatives ◽  
The Subject ◽  
The Stability

AbstractThe subject of this paper is the maximum principle and its application for investigating the stability and convergence of finite difference schemes. To some extent, this is a survey of the works on constructing and investigating certain new classes of monotone difference schemes. In this connection the maximum principle for the derivatives discussed in this paper is of fundamental importance. It is used as a basis for proving the coefficient stability of difference schemes in Banach spaces and the monotonicity of economical schemes of full approximation. New results on unconditional stability of monotone difference schemes with weights, conservative explicit-implicit schemes (staggered schemes), monotone schemes of second-order approximation in arbitrary domains, and monotone difference schemes for multidimensional elliptic equations with mixed derivatives are given.

scholarly journals A Splitting Scheme to Solve an Equation for Fractional Powers of Elliptic Operators

2016 ◽  
Vol 16 (1) ◽  
pp. 161-174 ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Dirichlet Boundary ◽  
Model Problem ◽  
Fractional Power ◽  
Dirichlet Boundary Conditions ◽  
Splitting Scheme ◽  
Fractional Powers ◽  
Splitting Schemes ◽  
Unconditionally Stable ◽  
Spatial Variables ◽  
Finite Difference Approximations

AbstractAn equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an auxiliary Cauchy problem for a pseudo-parabolic equation. Unconditionally stable vector-additive schemes (splitting schemes) are constructed. Numerical results for a model problem in a rectangle calculated using the splitting with respect to spatial variables are presented.

scholarly journals Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1486
Author(s):  
Binhu Xia ◽  
Yibao Li ◽  
Zhong Li
Keyword(s):  
Numerical Experiments ◽  
Direct Methods ◽  
Second Order ◽  
Splitting Scheme ◽  
Surface Mesh ◽  
Unconditionally Stable ◽  
Discrete Equations ◽  
Energy Stable

This paper describes temporally second-order unconditionally stable direct methods for Allen–Cahn and conservative Allen–Cahn equations on surfaces. The discretization is performed via a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation. We prove that the proposed schemes, which combine a linearly stabilized splitting scheme, are unconditionally energy-stable. The resulting system of discrete equations is linear and is simple to implement. Several numerical experiments are performed to demonstrate the performance of our proposed algorithm.

scholarly journals THE STABILITY CONDITIONS OF FINITE DIFFERENCE SCHEMES FOR SCHRÖDINGER, KURAMOTO‐TSZUKI AND HEAT EQUATIONS

1998 ◽  
Vol 3 (1) ◽  
pp. 177-194 ◽  
Author(s):  
M. Radžiūnas ◽  
F. Ivanauskas
Keyword(s):  
Finite Difference ◽  
Initial Boundary ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Heat Equations ◽  
Initial Boundary Value Problems ◽  
L2 Norm ◽  
The Stability ◽  
Conditions Of Stability ◽  
Initial Boundary Value

We consider various finite difference schemes for the first and the second initial‐boundary value problems for linear Kuramoto‐Tsuzuki, heat and Schrödinger equations in d‐dimensional case. Using spectral methods, we find the conditions of stability on initial data in the L2 norm.

THE FINITE DIFFERENCE SCHEME FOR MATHEMATICAL MODELING OF WOOD DRYING PROCESS

2001 ◽  
Vol 6 (1) ◽  
pp. 48-57 ◽  
Author(s):  
R. Čiegis ◽  
V. Starikovičius
Keyword(s):  
Mathematical Modeling ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Drying Process ◽  
Moisture Movement ◽  
Different Types ◽  
The Stability

This work discusses issues on the design of finite difference schemes for modeling the moisture movement process in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for different types of boundary conditions.

Stability, Accuracy, and Efficiency of Sequential Methods for Coupled Flow and Geomechanics

SPE Journal ◽  
2011 ◽  
Vol 16 (02) ◽  
pp. 249-262 ◽  
Author(s):  
J.. Kim ◽  
H.A.. A. Tchelepi ◽  
R.. Juanes
Keyword(s):  
Numerical Solutions ◽  
Stability Limit ◽  
Stability And Convergence ◽  
Coupled Flow ◽  
Unconditionally Stable ◽  
Von Neumann ◽  
Solution Strategies ◽  
Solution Methods ◽  
The Stability ◽  
Stability Limits

Summary We perform detailed stability and convergence analyses of sequential-implicit solution methods for coupled fluid flow and reservoir geomechanics. We analyze four different sequential-implicit solution strategies, where each subproblem (flow and mechanics) is solved implicitly: two schemes in which the mechanical problem is solved first—namely, the drained and undrained splits—and two schemes in which the flow problem is solved first—namely, the fixed-strain and fixed-stress splits. The von Neumann method is used to obtain the linear-stability criteria of the four sequential schemes, and numerical simulations are used to test the validity and sharpness of these criteria for representative problems. The analysis indicates that the drained and fixed-strain splits, which are commonly used, are conditionally stable and that the stability limits depend only on the strength of coupling between flow and mechanics and are independent of the timestep size. Therefore, the drained and fixed-strain schemes cannot be used when the coupling between flow and mechanics is strong. Moreover, numerical solutions obtained using the drained and fixed-strain sequential schemes suffer from oscillations, even when the stability limit is honored. For problems where the deformation may be plastic (nonlinear) in nature, the drained and fixed-strain sequential schemes become unstable when the system enters the plastic regime. On the other hand, the undrained and fixed-stress sequential schemes are unconditionally stable regardless of the coupling strength, and they do not suffer from oscillations. While both the undrained and fixed-stress schemes are unconditionally stable, for the cases investigated we found that the fixed-stress split converges more rapidly than the undrained split. On the basis of these findings, we strongly recommend the fixed-stress sequential-implicit method for modeling coupled flow and geomechanics in reservoirs.

scholarly journals Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1975
Author(s):  
M. Luísa Morgado ◽  
Magda Rebelo ◽  
Luís L. Ferrás
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Uniform Mesh ◽  
Numerical Schemes ◽  
The Stability ◽  
Distributed Order

In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.

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