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

ABSTRACTWe 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 O(Δx4+Δy4+Δt2) 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 O(Δx4+Δy4+Δt2(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.