Complete monotonicity-preserving numerical methods for time fractional ODEs
The time fractional ODEs are equivalent to convolutional Volterra integral equations with completely monotone kernels. We therefore introduce the concept of complete monotonicity-preserving (𝒞ℳ-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernels inherit the 𝒞ℳ property as the continuous equations. We prove that 𝒞ℳ-preserving schemes are at least A(π/2) stable and can preserve the monotonicity of solutions to scalar nonlinear autonomous fractional ODEs,
... both of which are novel. Significantly, by improving a result of Li and Liu (Quart. Appl. Math., 76(1):189-198, 2018), we show that the ℒ1 scheme is 𝒞ℳ-preserving, so that the ℒ1 scheme is at least A(π/2) stable, which is an improvement on stability analysis for ℒ1 scheme given in Jin, Lazarov and Zhou (IMA J. Numer. Analy. 36:197-221, 2016). The good signs of the coefficients for such class of schemes ensure the discrete fractional comparison principles, and allow us to establish the convergence in a unified framework when applied to time fractional sub-diffusion equations and fractional ODEs. The main tools in the analysis are a characterization of convolution inverses for completely monotone sequences and a characterization of completely monotone sequences using Pick functions due to Liu and Pego (Trans. Amer. Math. Soc. 368(12):8499-8518, 2016). The results for fractional ODEs are extended to 𝒞ℳ-preserving numerical methods for Volterra integral equations with general completely monotone kernels. Numerical examples are presented to illustrate the main theoretical results.