SURVEY PAPER AN APPROACH TO CONSTRUCT HIGHER ORDER TIME DISCRETISATION SCHEMES FOR TIME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS WITH NONSMOOTH DATA Neville J. Ford 1 , Yubin Yan 2 Dedicated to the 20th anniversary of FCAA and 65th anniversary of Professor Virginia Kiryakova Also, to the Memory of Christopher TH Baker (1939-2017) Abstract In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differ- ential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time dis- cretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singulari- ties of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich’s fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results. MSC 2010 : Primary 65M06; Secondary 65M12, 65M15, 26A33, 35R11 c  2017 Diogenes Co., Sofia pp. 1076–1105 , DOI: 10.1515/fca-2017-0058