On ninth order, explicit Numerov-type methods with constant coefficients. A new family of effectively nine stages, ninth-order hybrid explicit Numerov-type methods is presented for the solution of some special second order Initial Value Problem. After dealing with a reduced set of order conditions, we derive an optimal constant coefficients method along with a similar kind of method with reduced phase errors. We proceed with numerical tests using quadruple precision arithmetic on some well-known problems from the relevant literature. Finally, in the appendices, we list Mathematica packages implementing the corresponding algorithms.

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  1. Shokri, Ali; Mehdizadeh Khalsaraei, Mohammad: A new implicit high-order six-step singularly P-stable method for the numerical solution of Schrödinger equation (2021)
  2. Fang, Jie; Liu, Chenglian; Simos, T. E.; Famelis, I. Th.: Neural network solution of single-delay differential equations (2020)
  3. Khalsaraei, Mohammad Mehdizadeh; Shokri, Ali: An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems (2020)
  4. Khalsaraei, Mohammad Mehdizadeh; Shokri, Ali; Molayi, Maryam: The new class of multistep multiderivative hybrid methods for the numerical solution of chemical stiff systems of first order IVPs (2020)
  5. Konguetsof, A.: Algorithm for the development of families of numerical methods based on phase-lag Taylor series (2020)
  6. Medvedev, Maxim A.; Simos, T. E.; Tsitouras, Ch.: Explicit, eighth-order, four-step methods for solving (y^\prime\prime=f(x, y)) (2020)
  7. Simos, T. E.; Tsitouras, Ch.: Explicit, ninth order, two step methods for solving inhomogeneous linear problems (x”(t)= \Lambdax(t)+f(t)) (2020)
  8. Tsitouras, Ch.: Eighth order, phase-fitted, six-step methods for solving (y^\prime\prime=f(x,y)) (2020)
  9. Alolyan, Ibraheem; Simos, T. E.: A four-stages multistep fraught in phase method for quantum chemistry problems (2019)
  10. Alolyan, Ibraheem; Simos, T. E.: New multiple stages multistep method with best possible phase properties for second order initial/boundary value problems (2019)
  11. Chen, Zhong; Liu, Chenglian; Hsu, Chieh-Wen; Simos, T. E.: A new multistage multistep full in phase algorithm with optimized characteristics for problems in chemistry (2019)
  12. Hui, Fei; Simos, T. E.: New multistage two-step complete in phase scheme with improved properties for quantum chemistry problems (2019)
  13. Lin, Chialiang; Chen, Jwu Jenq; Simos, T. E.; Tsitouras, Ch.: Evolutionary derivation of sixth-order P-stable SDIRKN methods for the solution of PDEs with the method of lines (2019)
  14. Liu, Chenglian; Hsu, Chieh-Wen; Tsitouras, Ch.; Simos, T. E.: Hybrid Numerov-type methods with coefficients trained to perform better on classical orbits (2019)
  15. Lv, Jieyin; Simos, T. E.: A Runge-Kutta type crowded in phase algorithm for quantum chemistry problems (2019)
  16. Medvedev, Maxim A.; Simos, Theodore E.; Tsitouras, Charalampos: Low-order, P-stable, two-step methods for use with lax accuracies (2019)
  17. Medvedev, Maxim A.; Simos, Theodore E.; Tsitouras, Charalampos: Local interpolants for Numerov-type methods and their implementation in variable step schemes (2019)
  18. Qiu, Guo-Hua; Liu, Chenglian; Simos, T. E.: A new multistep method with optimized characteristics for initial and/or boundary value problems (2019)
  19. Qiu, Junlai; Huang, Junjie; Simos, T. E.: A perfect in phase FD algorithm for problems in quantum chemistry (2019)
  20. Tsitouras, Ch.: Explicit Runge-Kutta methods for starting integration of Lane-Emden problem (2019)

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