Sinc approximation of solution of heat equation with discontinuous initial condition. (English) Zbl 0833.65113
Bowers, K. L. (ed.) et al., Computation and control IV. Proceedings of the fourth Bozeman conference, Bozeman, MT, USA, August 3-9, 1994. Boston, MA: Birkhäuser. Prog. Syst. Control Theory. 20, 289-303 (1995).
The authors study the numerical solution of the one-dimensional heat equation with the initial data being discontinuous at the origin, using the heat kernel and the Sinc collocation method for convolution integrals [cf. F. Stenger, Collocating convolutions, Math. Comput. 64, No. 209, 211-235 (1995; Zbl 0828.65017)]. Stenger has shown that the method converges at an exponential rate.
In this paper the authors compare the following formulations: the first one based on the smoothing properties of the heat kernel is to apply the Sinc collocation algorithm presented in [loc. cit.]to the double integral for the forcing term ignoring the discontinuity in the initial data; the other one which ignores the smoothing properties is to break the double integral into two integrals, one over \(\mathbb{R}^- \times [0,T ]\) and the other over \(\mathbb{R}^+ \times [0,T ]\), such that only one integral is computed using a change of variable.
The numerical results in the paper show that for the considered problem, the first formulation is more efficient and more accurate than the second one. It is pointed out that the method presented in this paper is easily generalized to the case of finitely many jumps in the initial data.
For the entire collection see [Zbl 0819.00052].
In this paper the authors compare the following formulations: the first one based on the smoothing properties of the heat kernel is to apply the Sinc collocation algorithm presented in [loc. cit.]to the double integral for the forcing term ignoring the discontinuity in the initial data; the other one which ignores the smoothing properties is to break the double integral into two integrals, one over \(\mathbb{R}^- \times [0,T ]\) and the other over \(\mathbb{R}^+ \times [0,T ]\), such that only one integral is computed using a change of variable.
The numerical results in the paper show that for the considered problem, the first formulation is more efficient and more accurate than the second one. It is pointed out that the method presented in this paper is easily generalized to the case of finitely many jumps in the initial data.
For the entire collection see [Zbl 0819.00052].
Reviewer: S.Jiang (Bonn)
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
