The numerical solution of advection–diffusion problems using new cubic trigonometric B-splines approach

•A new cubic trigonometric B-spline method is presented for advection–diffusion problem.•Stability and order of convergence of the proposed numerical scheme is investigated.•Results show good agreement between numerical and exact solutions.•It stabilizes the solution of advection–diffusion equation which has the sharp gradients.•It is superior to redefined, cubic, exponential B-spline, MOCS, CGSG, TC, TG, CD6 due its accuracy.

A new cubic trigonometric B-spline collocation approach is developed for the numerical solution of the advection–diffusion equation with Dirichlet and Neumann's type boundary conditions. The approach is based on the usual finite difference scheme to discretize the time derivative while a cubic trigonometric B-spline is utilized as an interpolation function in the space dimension with the help of θ-weighted scheme. The present scheme stabilizes the oscillations that are normally displayed by the approximate solution of the transient advective–diffusive equation in the locality of sharp gradients produced by transient loads and boundary layers. The scheme is shown to be stable and the accuracy of the scheme is tested by application to various test problems. The proposed approach is numerically verified to second order and shown to work for the Péclet number ≤ 5.