[Paper Review] Preconditioned Implicit-Exponential (IMEXP) Time Integrators for Stiff Differential Equations

Vũ Thái Luân, Mayya Tokman+1 more|arXiv (Cornell University)|May 10, 2016

Numerical methods for differential equations34 references3 citations

TL;DR

This paper introduces two new classes of time integrators—Implicit-Exponential (IMEXP) and Hybrid Exponential (HImExp)—for stiff differential equations, where both linear (L) and nonlinear (N) operators are stiff. By combining implicit treatment of L with exponential integration of N and enabling use of any preconditioner, the methods achieve superior stability and efficiency, especially for highly stiff problems, with computational savings up to 87% compared to 2-sBDF when preconditioned.

ABSTRACT

We propose two new classes of time integrators for stiff DEs: the implicit-explicit exponential (IMEXP) and the hybrid exponential methods. In contrast to the existing exponential schemes, the new methods offer significant computational advantages when used with preconditioners. Any preconditioner can be used with any of these new schemes. This leads to a broader applicability of exponential methods. The proof of stability and convergence of these integrators and numerical demonstration of their efficiency are presented.

Motivation & Objective

• To address the computational inefficiency of existing IMEX and exponential integrators when both linear and nonlinear operators in stiff ODEs are stiff.
• To develop new time integrators that can effectively leverage existing preconditioners for both stiff operators, overcoming limitations of standard exponential methods.
• To provide a stable, convergent, and computationally efficient alternative to IMEX schemes for problems with stiff nonlinearities.
• To demonstrate through theory and numerical experiments that the new methods outperform standard 2-sBDF and IMEX schemes, especially under high stiffness and with preconditioning.

Proposed method

• Proposes two new classes of time integrators: Implicit-Exponential (IMEXP) and Hybrid Exponential (HImExp), designed to treat stiff linear and nonlinear terms separately.
• Uses exponential integration for the nonlinear term N(u(t)) while treating the linear term L(u(t)) implicitly, enabling better stability for stiff nonlinearities.
• Introduces a novel formulation that allows any existing preconditioner to be applied to either L or N, enhancing flexibility and computational efficiency.
• Derives stiffly accurate schemes and establishes convergence and stability theory for the new methods, ensuring robustness for long-time integration.
• Employs Krylov subspace methods for matrix exponential computations, reducing the cost of exponential integrator evaluation.
• Validates the approach numerically using 2D Allen-Cahn and Schnakenberg reaction-diffusion equations with varying stiffness parameters and preconditioning.

Experimental results

Research questions

• RQ1 Can exponential integrators be effectively extended to handle stiff nonlinear operators when standard IMEX methods fail due to explicit treatment of N?
• RQ2 How can preconditioners be systematically incorporated into exponential integrators to improve efficiency for both linear and nonlinear stiff terms?
• RQ3 What is the stability and convergence behavior of new IMEXP and HImExp schemes when both L and N are stiff?
• RQ4 Can the new methods achieve significant computational savings over 2-sBDF and other IMEX schemes, especially under high stiffness and preconditioning?
• RQ5 How does the performance of the new schemes scale with increasing stiffness in the nonlinear term N(u(t))?

Key findings

• The HImExp2J scheme consistently outperforms 2-sBDF and other IMEXP methods across all test problems, achieving up to 87% reduction in CPU time for the most stiff case.
• For the 2D Schnakenberg system with γ = 5×10⁴, the preconditioned HImExp2J method reduced CPU time to 20% of 2-sBDF at 10⁻³ accuracy, compared to 70% for non-preconditioned.
• The computational advantage of HImExp2J grows with increasing stiffness: at 10⁻³ tolerance, savings were 87% for the stiffest Schnakenberg case with preconditioning.
• For the Allen-Cahn equation, HImExp2J achieved a 55% reduction in CPU time at 10⁻² accuracy with ε = 0.02, even without preconditioning.
• The non-preconditioned HImExp2J method was faster than 2-sBDF at 10⁻³ accuracy for ε = 0.005, indicating improved stability allows larger time steps.
• Numerical results confirm theoretical predictions: HImExp2J maintains superior performance across varying stiffness levels and preconditioning, validating its robustness and scalability.