Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.

[1]  A. Gulin,et al.  Stability of a Nonlocal Two-Dimensional Finite-Difference Problem , 2001 .

[2]  N. Borovykh.  Stability in the numerical solution of the heat equation with nonlocal boundary conditions , 2002 .

[3]  F. Ivanauskas,et al.  An Explicit Solution of a Parabolic Equation with Nonlocal Boundary Conditions , 2005 .

[4]  Olga Štikonienė,et al.  A Monotonic Finite-Difference Scheme for a Parabolic Problem with Nonlocal Conditions , 2002 .

[5]  G. Kairyte,et al.  Alternating direction method for a two‐dimensional parabolic equation with a nonlocal boundary condition , 2007 .

[6]  Gunnar Ekolin,et al.  Finite difference methods for a nonlocal boundary value problem for the heat equation , 1991 .

[7]  Yanping Lin,et al.  The solution of the diffusion equation in two space variables subject to the specification of mass , 1993 .

[8]  Yanping Lin,et al.  FINITE DIFFERENCE APPROXIMATIONS FOR A CLASS OF NON-LOCAL PARABOLIC EQUATIONS , 1997 .

[10]  Abdesslam Boutayeb,et al.  Global Extrapolations Of Numerical Methods For Solving A Parabolic Problem With Non Local Boundary Conditions , 2003, Int. J. Comput. Math..

[11]  Baruch Cahlon,et al.  Stepwise stability for the heat equation with a nonlocal constraint , 1995 .

[12]  Juan C. López-Marcos,et al.  Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions , 1996, Adv. Comput. Math..

[13]  V. Makarov,et al.  Eigenvalue Problem for the Second Order Differential Equation with Nonlocal Conditions , 2006 .

[14]  UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports , 1996 .

[15]  R. Čiegis.  Economical Difference Schemes for the Solution of a Two-Dimensional Parabolic Problem with an Integral Condition , 2005 .

[16]  L. Shampine,et al.  A BVP Solver that Controls Residual and Error 1 , 2008 .

[17]  M. Dehghan.  Efficient techniques for the second-order parabolic equation subject to nonlocal specifications , 2005 .

[18]  M. Sapagovas.  The Eigenvalues of Some Problems with a Nonlocal Condition , 2002 .

[19]  William Alan Day,et al.  Extensions of a property of the heat equation to linear thermoelasticity and other theories , 1982 .

[20]  Mehdi Dehghan,et al.  Locally explicit schemes for three-dimensional diffusion with a non-local boundary specification , 2003, Appl. Math. Comput..

[21]  Hong Du,et al.  The solution of a parabolic differential equation with non-local boundary conditions in the reproducing kernel space , 2008, Appl. Math. Comput..

[22]  M. Sapagovas,et al.  On the Structure of the Spectrum of a Differential Operator with a Nonlocal Condition , 2005 .