Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem
Chaobao Huang, Xiaohui Liu, Xiangyun Meng, Martin Stynes
Abstract
Abstract An initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered. Its spatial domain is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>d</m:mi> </m:msup> </m:math> {(0,1)^{d}} for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>3</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {d\in\{1,2,3\}} . This problem is a generalisation of the problem considered by Stynes, O’Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {d=1} and only one fractional time derivative was present. A priori bounds on the derivatives of the unknown solution are derived. A finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time. Stability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way. Numerical results supporting our theoretical results are provided.