High order numerical algorithms based on biquadratic spline collocation for two-dimensional parabolic partial differential equations

ABSTRACT

We report a new algorithm for solving linear parabolic partial differential equations in two space dimension. The algorithm employs optimal biquadratic spline collocation for space discretization and modified trapezoidal rule for time discretization. We need to solve a block tridiagonal linear system at each time step, and obtain an approximate solution with error $\mathcal{O}(\mathrm{\Delta }{x}^4+\mathrm{\Delta }{y}^4+\mathrm{\Delta }{t}^2)$ at space-time grid points. We analyse the stability of the new algorithm, and present a stability enhanced variant. Moreover, we give an acceleration strategy based on spectral deferred correction, and the theoretical accuracy can be increased to $\mathcal{O}(\mathrm{\Delta }{x}^4+\mathrm{\Delta }{y}^4+\mathrm{\Delta }{t}^{2(k+1)})$, where k is the number of correction loops. We also analyse the stability for the accelerated algorithms. Numerical experiments are attached to demonstrate the effectiveness of the new algorithms.

KEYWORDS:

• Biquadratic spline collocation
• Crank–Nicolson
• convergence
• stability
• spectral deferred correction

2010 AMS(MOS) SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M70

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [numbers 11401589 and 11571367], the Natural Science Foundation of Shandong Province [number ZR2013AL018] and the Fundamental Research Funds for the Central Universities [numbers 18CX02049A and 17CX02066].