scholarly journals A Family of Finite-Difference Schemes with Discrete Transparent Boundary Conditions for a Parabolic Equation on the Half-Axis

2013 ◽  
Vol 13 (2) ◽  
pp. 119-138
Author(s):  
Alexander Zlotnik ◽  
Natalya Koltsova
Keyword(s):  
Parabolic Equation ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Transparent Boundary Conditions ◽  
Half Axis ◽  
Two Parameters

Abstract. An initial-boundary value problem for the 1D self-adjoint parabolic equation on the half-axis is solved. We study a broad family of two-level finite-difference schemes with two parameters related to averages both in time and space. Stability in two norms is proved by the energy method. Also discrete transparent boundary conditions are rigorously derived for schemes by applying the method of reproducing functions. Results of numerical experiments are included as well.

Stability of Finite-difference Schemes for Semilinear Multidimensional Parabolic Equations

2012 ◽  
Vol 12 (3) ◽  
pp. 289-305 ◽  
Author(s):  
Bosko Jovanovic ◽  
Magdalena Lapinska-Chrzczonowicz ◽  
Aleh Matus ◽  
Piotr Matus
Keyword(s):  
Finite Difference ◽  
Parabolic Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Nonlinear Odes ◽  
The Stability ◽  
Initial Boundary Value

Abstract Abstract — We have studied the stability of finite-difference schemes approximating initial-boundary value problem (IBVP) for multidimensional parabolic equations with a nonlinear source of a power type. We have obtained simple sufficient input data conditions, in which the solutions of differential and difference problems are globally bounded for all t. It is shown that if these conditions are not satisfied, then the solution can blow-up (go to infinity) in finite time. The lower bound of the blow-up time has been determined. The stability of the difference solution has been proven. In all cases, we used the method of energy inequalities based on the application of the Chaplygin comparison theorem for nonlinear ODEs, Bihari-type inequalities and their discrete analogs.

Remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis

2016 ◽  
Vol 31 (1) ◽  
Author(s):  
Alexander Zlotnik ◽  
Ilya Zlotnik
Keyword(s):  
Finite Difference ◽  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Finite Difference Schemes ◽  
Transparent Boundary Conditions ◽  
Space Step ◽  
Half Axis ◽  
Generalized Time

AbstractWe consider the generalized time-dependent Schrödinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step

Two Finite Difference Schemes for Multi-Dimensional Fractional Wave Equations with Weakly Singular Solutions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jinye Shen ◽  
Martin Stynes ◽  
Zhi-Zhong Sun
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Mesh Size ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Initial Time ◽  
Finite Difference Schemes

Abstract A time-fractional initial-boundary value problem of wave type is considered, where the spatial domain is ( 0 , 1 ) d (0,1)^{d} for some d ∈ { 1 , 2 , 3 } d\in\{1,2,3\} . Regularity of the solution 𝑢 is discussed in detail. Typical solutions have a weak singularity at the initial time t = 0 t=0 : while 𝑢 and u t u_{t} are continuous at t = 0 t=0 , the second-order derivative u t ⁢ t u_{tt} blows up at t = 0 t=0 . To solve the problem numerically, a finite difference scheme is used on a mesh that is graded in time and uniform in space with the same mesh size ℎ in each coordinate direction. This scheme is generated through order reduction: one rewrites the differential equation as a system of two equations using the new variable v := u t v:=u_{t} ; then one uses a modified L1 scheme of Crank–Nicolson type for the driving equation. A fast variant of this finite difference scheme is also considered, using a sum-of-exponentials (SOE) approximation for the kernel function in the Caputo derivative. The stability and convergence of both difference schemes are analysed in detail. At each time level, the system of linear equations generated by the difference schemes is solved by a fast Poisson solver, thereby taking advantage of the fast difference scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of both numerical methods.

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