“…In order to solve Equation (3.3) effectively, we consider two specific classes of problems:- When the diffusion coefficient ξ ( x , t ) ≡ ξ , the coefficient matrix of Equation (3.3) will be a time‐independent Toeplitz matrix, that is, ℳ j + 1 = ℳ; then we can compute its matrix inverse via the Gohberg–Semencul formula (GSF) [13] using only its first and last columns. Such a strategy does not need to call the preconditioned Krylov subspace solvers at each time level 0 ≤ j ≤ M − 1, and the solution at each time level (i.e., ℳ −1 u j + 1 ) can be calculated via about six FFTs, thus saving considerable computational cost; refer to [14, 18, 32, 60] for detail.
- When the diffusion coefficient is just a function related to both x and t , that is, ξ ( x , t ), the coefficient matrix of Equation (3.3) becomes the sum of a scalar matrix and of a diagonal‐multiply‐Toeplitz matrix, which is time‐dependent. In this case, Equation (3.3) has to be solved via a preconditioned Krylov subspace solver at each time level j .
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