Convergence analysis of adi orthogonal spline collocation without perturbation terms

For the heat equation on a rectangle and nonzero Dirichlet boundary conditions, we consider an ADI orthogonal spline collocation method without perturbation terms, to specify boundary values of intermediate solutions at half time levels on the vertical sides of the rectangle. We show that, at each time level, the method has optimal convergence rate in the L² norm in space. Numerical results for splines of orders 4, 5, 6 confirm our theoretical convergence rates and demonstrate suboptimal convergence rates in the H¹ norm. We also demonstrate numerically that the scheme without the perturbation terms is applicable to variable coefficient problems yielding the same convergence rates obtained for the heat equation.