Derivation and error analysis of three-level linearized difference schemes for solving the Burgers-Fisher equation

The main scope of this paper is to develop and analyse three-level linearized difference schemes for solving the classical Burgers-Fisher equation. For the Dirichlet boundary value problem, the first three-level linearized difference scheme is second-order accurate in both time and space. It is able to obey a discrete conservative law. By the discrete energy argument and induction, it is rigorously proved to be uniquely solvable and unconditionally convergent. Furthermore, with the purpose of improving the numerical accuracy in space, another three-level linearized compact difference scheme is then established together with some investigation on its discrete conservation law, unique solvability and unconditional convergence of order two in time and four in space. The coupled nonlinearity of Burgers' type and Fisher type is intensively treated via the order of reduction and Taylor expansion. In addition, extensions to the problem subject to the periodic boundary condition (PBC) are involved. To the best of our knowledge, this is a rare work to carefully design and rigorously analyse the high-order linearized difference schemes for solving the Burgers-Fisher equation. Numerical tests are provided to support the theoretical results and to verify the computational efficiency.