Efficient Krylov-based exponential time differencing method in application to 3D advection-diffusion-reaction systems

The number of ordinary differential equations generally increases exponentially as the partial differential equation is posed on a domain with more dimensions. This is, of course, the curse of dimensionality for exponential time differencing methods. The computational challenge in applying exponential time differencing methods for solving partial differential equations in high spatial dimensions is how to compute the matrix exponential functions for very large matrices accurately and efficiently. In this paper, our main aim is to design a Krylov subspace approximation-based locally extrapolated exponential time differencing method and compare its performance in terms of accuracy and efficiency to the already available method in the literature for solving a three-dimensional nonlinear advection-diffusion-reaction systems. The fundamental idea of the proposed method is to compute only the action of the matrix exponential on a given state vector instead of computing the matrix exponential itself, and then multiplying it with given vector. The stability and local truncation error of the proposed method have been examined. Calculation of local truncation error and empirical convergence analysis indicate the proposed method is second-order accurate in time. The performance and reliability of this novel method have been investigated by testing it on systems of the three-dimensional nonlinear advection-diffusion-reaction equations and three-dimensional viscous nonlinear Burgers' equation.