Sixth-order explicit one-step methods for stiff ODEs via hybrid deferred correction involving RK2 and RK4: Application to reaction-diffusion equations

In this paper, the fourth-order explicit Runge-Kutta method (RK4) is used to make a Deferred Correction (DC) on the explicit midpoint rule, resulting in an explicit one-step method of order six of accuracy, denoted DC6RK2/4. Convergence and order of accuracy of DC6RK2/4 are proven through a deferred correction condition satisfied by the RK4. The region of absolute stability of this method contains that of a RK6 and is tangent to the region [-5.626,0[x[-4.730,4.730] of the complex plane, containing a significant part of the imaginary axis. Numerical experiments with standard test problems for stiff systems of ODEs show that DC6RK2/4 performs well on problems regarding strong non-linearity and long-term integration, and this method does not require extremely small time steps for accurate numerical solutions of stiff problems. Moreover, this method is better than standard implicit methods like the Backward Differentiation Formulae and the DC methods for the implicit midpoint rule on stiff problems for which Jacobian matrices along the solution curve have complex eigenvalues where imaginary parts have larger magnitudes than real parts. An application of DC6RK2/4 to a class of test problems for reaction-diffusion equations in one dimensional is also carried out.

💡 Research Summary

The paper introduces a novel sixth‑order explicit one‑step method, denoted DC6RK2/4, designed for stiff ordinary differential equations (ODEs) and reaction‑diffusion problems. The construction starts from the explicit midpoint rule (a second‑order Runge‑Kutta method, RK2) and applies a deferred‑correction (DC) procedure using the classical fourth‑order Runge‑Kutta method (RK4). By approximating the local truncation errors of the midpoint rule with solutions obtained from RK4 on five sub‑steps within each main step, two correction constants (aₙ and bₙ) are formed. The resulting update formulas retain the simple structure of the midpoint method but incorporate these constants, yielding a scheme that attains global accuracy of order six while requiring only 21 function evaluations per step (5 RK4 sub‑steps plus the two RK2 evaluations).

A central theoretical contribution is the formulation of a Deferred Correction Condition (DCC) for RK4. The DCC requires that forward differences of the RK4 solution satisfy ‖D⁺ᵐ(u₄,n – u(tₙ))‖ ≤ C·k⁴ for all relevant m, which the authors prove under the assumption that the right‑hand side F is sufficiently smooth (C⁶) and the exact solution u belongs to C⁷. By induction on the difference order, they show that the DCC holds for RK4, guaranteeing that the hybrid correction inherits the sixth‑order convergence.

Stability analysis is performed on the linear test equation y′ = λy. The amplification factor of DC6RK2/4 is derived, and the absolute stability region is shown to contain the entire region of the classical sixth‑order Runge‑Kutta (RK6). Notably, the region extends to the interval