[Paper Review] A fourth-order compact solver for fractional-in-time fourth-order diffusion equations

Jialing Zhong, Hong-lin Liao+2 more|arXiv (Cornell University)|Jul 3, 2019

Fractional Differential Equations Solutions26 references4 citations

TL;DR

This paper proposes a fourth-order compact finite difference scheme for time-fractional fourth-order subdiffusion equations with first Dirichlet boundary conditions. By reformulating the problem into a second-order system and employing an averaged compact operator for spatial approximation, the method achieves fourth-order spatial accuracy and handles initial singularity via a graded nonuniform L1 formula for the Caputo derivative, with rigorous stability and convergence analysis using discrete fractional Grönwall inequality and error convolution structure, confirming optimal temporal convergence of order $ O(\tau^{\min\{\gamma\sigma,2-\alpha\}}) $.

ABSTRACT

A fourth-order compact scheme is proposed for a fourth-order subdiffusion equation with the first Dirichlet boundary conditions. The fourth-order problem is firstly reduced into a couple of spatially second-order system and we use an averaged operator to construct a fourth-order spatial approximation. This averaged operator is compact since it involves only two grid points for the derivative boundary conditions. The L1 formula on irregular mesh is considered for the Caputo fractional derivative, so we can resolve the initial singularity of solution by putting more grid points near the initial time. The stability and convergence are established by using three theoretical tools: a complementary discrete convolution kernel, a discrete fractional Gronwall inequality and an error convolution structure. Some numerical experiments are reported to demonstrate the accuracy and efficiency of our method.

Motivation & Objective

• To develop a high-order numerical method for time-fractional fourth-order subdiffusion equations with first Dirichlet boundary conditions.
• To address the initial singularity in the solution caused by the Caputo fractional derivative through a graded, nonuniform time mesh.
• To achieve fourth-order spatial accuracy using a compact finite difference scheme based on an averaged operator.
• To establish rigorous stability and convergence of the scheme using theoretical tools such as the discrete fractional Grönwall inequality and error convolution structure.
• To validate the theoretical findings through extensive numerical experiments with varying fractional orders and mesh parameters.

Proposed method

• The fourth-order subdiffusion equation is reformulated into a coupled system of second-order equations using the auxiliary variable $ v = \partial_x^2 u $.
• A compact averaging operator is constructed to achieve fourth-order spatial accuracy while maintaining compact stencil involving only two grid points for boundary conditions.
• A nonuniform L1 formula is applied to the Caputo fractional derivative on a graded mesh $ t_k = (k/N)^\gamma T $, with $ \gamma \geq 1 $, to resolve initial singularity.
• Stability and convergence are proven using three key tools: a complementary discrete convolution kernel, a discrete fractional Grönwall inequality, and an error convolution structure.
• The method is implemented in both space and time with a uniform spatial mesh and graded temporal mesh, and convergence rates are computed via $ \text{Order}(M) \approx \log_2(e(M,N)/e(2M,N)) $ and $ \text{Order}(N) \approx \log_2(e(M,N)/e(M,2N)) $.
• Numerical experiments are conducted across multiple scenarios with varying $ \alpha $, $ \sigma $, $ \gamma $, and mesh sizes to validate accuracy and efficiency.

Experimental results

Research questions

• RQ1 Can a fourth-order compact scheme be constructed for time-fractional fourth-order subdiffusion equations with first Dirichlet boundary conditions?
• RQ2 How can the initial singularity in the solution be effectively resolved using a nonuniform time mesh?
• RQ3 What is the optimal temporal convergence rate of the scheme, and how does it depend on the grading parameter $ \gamma $ and solution regularity parameter $ \sigma $?
• RQ4 Can the stability and convergence of the scheme be rigorously proven using discrete fractional Grönwall inequality and error convolution structure?
• RQ5 Does the scheme achieve fourth-order spatial accuracy and optimal temporal convergence under varying fractional orders and mesh parameters?

Key findings

• The scheme achieves fourth-order spatial accuracy, confirmed by numerical experiments showing spatial convergence orders close to 4.00 across all tested cases.
• Temporal convergence rates match the theoretical prediction $ O(\tau^{\min\{\gamma\sigma,2-\alpha\}}) $, with optimal $ O(\tau^{2-\alpha}) $ observed when $ \gamma > (2-\alpha)/\sigma $.
• For uniform mesh ($ \gamma = 1 $), the temporal convergence order is $ O(\tau^{\sigma}) $, consistent with theoretical analysis and observed in Table 3.
• The scheme maintains stability and convergence even for high-regularity solutions with $ \sigma = 1.9 $, achieving temporal order $ \approx 1.10 $ for $ \alpha = 0.9 $.
• Numerical results in Tables 4–6 confirm that the $ L^\infty $ error estimate $ \|u - U^n\|_{\infty} = O(\tau^{\min\{\gamma\sigma,2-\alpha\}}) $ is sharp, with observed orders matching theoretical bounds.
• The method efficiently resolves initial layer behavior through graded meshing, as evidenced by consistent convergence rates even with strong initial singularities.