Numerical solution of a non-local boundary value problem with Neumann’s boundary conditions. (English) Zbl 1014.65072
Summary: Several second-order finite difference schemes are discussed for solving a non-local boundary value problem for two-dimensional diffusion equation with Neumann’s boundary conditions. While sharing some common features with the one-dimensional models, the solution of two-dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of one-dimensional case. Using a suitable transformation the solution of this problem is equivalent to the solution of two other problems. The former which is a one-dimensional non-local boundary value problem gives the value \(\mu\) of through using the unconditionally stable standard implicit (3,1) backward time centred space (denoted BTCS) scheme.
Using this result the second problem will be changed to a classical two-dimensional diffusion equation with Neumann’s boundary conditions which will be solved numerically by using two unconditionally stable fully implicit finite difference schemes, or using two conditionally stable fully explicit finite difference techniques. For each method investigated the modified equivalent partial differential equation is employed which permits the order of accuracy of the numerical techniques to be determined. The results of a numerical example for all finite difference schemes discussed in this paper are given and computation times are presented.
Using this result the second problem will be changed to a classical two-dimensional diffusion equation with Neumann’s boundary conditions which will be solved numerically by using two unconditionally stable fully implicit finite difference schemes, or using two conditionally stable fully explicit finite difference techniques. For each method investigated the modified equivalent partial differential equation is employed which permits the order of accuracy of the numerical techniques to be determined. The results of a numerical example for all finite difference schemes discussed in this paper are given and computation times are presented.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
Keywords:
finite difference methods; Neumann’s boundary conditions; parabolic partial differential equations; non-local boundary value problems; two-dimensional heat equation; central processor times; numerical exampleReferences:
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