A fourth-order accurate, three-point compact approximation of the boundary gradient, for electrochemical kinetic simulations by the extended Numerov method

The fourth-order accurate, three-point compact (extended Numerov) finite-difference scheme of Chawla [J. Inst. Math. Appl. 22 (1978) 89] has been recently found superior (in terms of accuracy and efficiency) to the conventional second-order accurate spatial discretisation commonly used in electrochemical kinetic simulations. However, the two-point compact boundary gradient approximation, accompanying the scheme, is difficult to apply in the case of time-dependent kinetic partial differential equations, because it introduces unwanted second temporal derivatives into calculations. The conventional five-point gradient formula is free from this drawback, but it is also not very convenient, owing to the locally increased bandwidth of the matrix of linear equations arising from the spatio-temporal discretisation. A new three-point compact boundary gradient approximation derived in this work, avoids the above inconveniences and economically re-uses expressions utilised by the extended Numerov discretisation. The fourth-order accuracy of the new approximation is proven theoretically and verified in computational experiments performed for examples of kinetic models.