Convergence analysis of compact difference schemes for diffusion equation with nonlocal boundary conditions

•Difference schemes for solving the diffusion equation with nonlocal boundary conditions are considered.•Compact difference is used and the integrals are approximated by the composite Simpson scheme.•Convergence of the two compact schemes are derived using the energy method.

Compact difference schemes for solving the diffusion equation with nonlocal boundary conditions are considered in this paper. Fourth-order compact difference is used to approximate the second order spatial derivative, and the integrals in the boundary conditions are approximated by the composite Simpson quadrature formula. The backward Euler and Crank–Nicolson schemes are presented as the fully discrete schemes. Error estimates in the discrete h1 and l∞ norms are given by the energy method, showing both schemes are fourth-order accurate in space, and they have first-order and second-order accuracy in time, respectively. Numerical results are provided to confirm the theoretical results.