The Construction of High-Order Robust Theta Methods with Applications in Subdiffusion Models

Abstract

An exponential-type function was discovered to transform known difference formulas by involving a shifted parameter θ to approximate fractional calculus operators. In contrast to the known θ methods obtained by polynomial-type transformations, our exponential-type θ methods take the advantage of the fact that they have no restrictions in theory on the range of θ such that the resultant scheme is asymptotically stable. As an application to investigate the subdiffusion problem, the second-order fractional backward difference formula is transformed, and correction terms are designed to maintain the optimal second-order accuracy in time. The obtained exponential-type scheme is robust in that it is accurate even for very small α and can naturally resolve the initial singularity provided θ=−12, both of which are demonstrated rigorously. All theoretical results are confirmed by extensive numerical tests.

Keywords

• theta methods
• subdiffusion problem
• fractional calculus
• backward difference formula
• convolution quadrature