**A. S. Hendy 1,2 and R. H. De Staelen 3,4,\***


Received: 31 August 2020; Accepted: 23 September 2020; Published: 03 October 2020

**Abstract:** In this paper, we introduce a high order numerical approximation method for convection diffusion wave equations armed with a multiterm time fractional Caputo operator and a nonlinear fixed time delay. A temporal second-order scheme which is behaving linearly is derived and analyzed for the problem under consideration based on a combination of the formula of *L*<sup>2</sup> − 1*<sup>σ</sup>* and the order reduction technique. By means of the discrete energy method, convergence and stability of the proposed compact difference scheme are estimated unconditionally. A numerical example is provided to illustrate the theoretical results.

**Keywords:** fractional convection diffusion-wave equations; compact difference scheme; nonlinear delay; spatial variable coefficients; convergence and stability

**MSC:** 65M06; 35K15; 35K55; 35K57
