Collection of extended stability Runge-Kutta schemes

A curated set of third- and fourth-order Extended-Stability Runge–Kutta (ESRK) methods with large real-axis stability regions—ready to use in ODE/PDE solvers and Neural ODEs. Scope. This repo focuses on practical, implementation-ready schemes: tabulated coefficients, stability diagnostics, and small utilities to verify order conditions and plot stability polynomials.

Third Order Schemes

ESRK(15,3) — Extended-Stability Runge–Kutta Scheme

Order: 3
Stages: 15
Type: Explicit RK (strictly lower-triangular A)
Family: ESRK — Extended-Stability Runge–Kutta
Design goal: Maximize the real-axis stability interval while retaining 3rd-order accuracy. Scheme: In the esrk(15,3) file

🔑 Scheme Summary

Property	Value
Name	ESRK(15,3)
Family	Extended-Stability Runge–Kutta (ESRK)
Order	3
Stages	15
Type	Explicit RK (lower-triangular (A))
Stability polynomial degree	15
Real-axis stability radius )	≈ 71.77823812058196,

ESRK(21,3) — Extended-Stability Runge–Kutta Scheme

Order: 3

Stages: 21

Type: Explicit RK (strictly lower-triangular A)

Family: ESRK — Extended-Stability Runge–Kutta

Design goal: Maximize the real-axis stability interval while retaining 3rd-order accuracy.

Scheme: In the esrk(21,3) file

🔑 Scheme Summary

Property	Value
Name	ESRK(21,3)
Family	Extended-Stability Runge–Kutta (ESRK)
Order	3
Stages	21
Type	Explicit RK (lower-triangular (A))
Stability polynomial degree	21
Real-axis stability radius )	≈ 141.49,

---

Fourth Order Schemes

ESRK(16,4) — Extended-Stability Runge–Kutta Scheme

Order: 4
Stages: 16
Type: Explicit RK (strictly lower-triangular A)
Family: ESRK — Extended-Stability Runge–Kutta
Design goal: Maximize the real-axis stability interval while retaining 4th-order accuracy. Scheme: In the esrk(16,4) file

🔑 Scheme Summary

Property	Value
Name	ESRK(16,4)
Family	Extended-Stability Runge–Kutta (ESRK)
Order	4
Stages	16
Type	Explicit RK (lower-triangular (A))
Stability polynomial degree	16
Real-axis stability radius )	≈ 58.656618727743464,

---

ESRK(16,4) — Extended-Stability Runge–Kutta Scheme + embeddings

🔑 Scheme Summary

Property	Value
Name	ESRK(16,4)
Family	Extended-Stability Runge–Kutta (ESRK)
Order	4
Stages	16
Type	Explicit RK (lower-triangular (A))
Stability polynomial degree	16
Real-axis stability radius )	≈ 58.656618727743464,
Embedded ratio (R_3/R_4)**	≈ 0.33
dev from main	4%