Two new classes of exponential Runge–Kutta integrators for efficiently solving stiff systems or highly oscillatory problems

References

• H. Berland, B. Owren, and B. Skaflestad, B-series and order conditions for exponential integrators, SIAM J. Numer. Anal. 43 (2005), pp. 1715–1727.
   Web of Science ®Google Scholar
• H. Berland, B. Skaflestad, and W. Wright, EXPINT–A MATLAB package for exponential integrators, ACM Trans. Math. Softw. 33 (2007), pp. 4.
   Web of Science ®Google Scholar
• G. Beylkin, J.M. Kelser, and L. Vozovol, A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comput. Phys. 147 (1998), pp. 362–387.
   Web of Science ®Google Scholar
• A. Bhatt and B.E. Moore, Structure-preserving exponential Runge–Kutta methods, SIAM J. Sci. Comput. 39 (2017), pp. A593–A612.
   Web of Science ®Google Scholar
• J.C. Butcher, Numerical Methods for Ordinary Diffrential Equations, 2nd ed., John Wiley & Sons, Ltd, West Sussex, England 2008.
   Google Scholar
• E. Celledoni, D. Cohen, and B. Owren, Symmetric exponential integrators with an application to the cubic Schröinger equation, Found. Comput. Math. 8 (2008), pp. 303–317.
   Web of Science ®Google Scholar
• J.B. Chen and M.Z. Qin, Multisymplectic Fourier pseudospectral method for the nonlinear Schrödinger equation, Electron. Trans. Numer. Anal. 12 (2001), pp. 193–204.
   Google Scholar
• S. Cox and P. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002), pp. 430–455.
   Web of Science ®Google Scholar
• G.G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), pp. 27–43.
   Google Scholar
• G. Dimarco and L. Pareschi, Exponential Runge–Kutta methods for stiff kinetic equations, SIAM J. Numer. Anal. 49 (2011), pp. 2057–2077.
   Web of Science ®Google Scholar
• G. Dujardin, Exponential Runge–Kutta methods for the Schröinger equation, Appl. Numer. Math. 59 (2009), pp. 1839–1857.
   Web of Science ®Google Scholar
• M. Hochbruck and C. Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal. 34 (1997), pp. 1911–1925.
   Web of Science ®Google Scholar
• M. Hochbruck, C. Lubich, and H. Selhofer, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput. 19 (1998), pp. 1552–1574.
   Web of Science ®Google Scholar
• M. Hochbruck and A. Ostermann, Explicit exponential Runge–Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal. 43 (2005), pp. 1069–1090.
   Web of Science ®Google Scholar
• M. Hochbruck and A. Ostermann, Exponential Runge–Kutta methods for parabolic problems, Appl. Numer. Math. 53 (2005), pp. 323–339.
   Web of Science ®Google Scholar
• M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer. 19 (2010), pp. 209–286.
   Google Scholar
• A.-K. Kassam and L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput. 26 (2005), pp. 1214–1233.
   Web of Science ®Google Scholar
• J.D. Lawson, Generalized Runge–Kutta processes for stable systems with large Lipschitz constants, SIAM J. Numer. Anal. 4 (1967), pp. 372–380.
   Google Scholar
• Y.W. Li and X. Wu, Exponential integrators preserving first integrals or Lyapunov functions for conservative or dissipative systems, SIAM J. Sci. Comput. 38 (2016), pp. A1876–A1895.
   Web of Science ®Google Scholar
• R.I. Mclachlan, G.R.W. Quispel, and N. Robidoux, A unified approach to Hamiltonian systems, poisson systems, gradient systems, and systems with Lyapunov functions or first integrals, Phys. Rev. Lett. 81 (1998), pp. 2399–2403.
   Web of Science ®Google Scholar
• L. Mei, L. Huang, and X. Wu, Energy-preserving continuous-stage exponential Runge–Kutta integrators for efficiently solving Hamiltonian systems, SIAM J. Sci. Comput. 44 (2022), pp. A1092–A1115.
   Web of Science ®Google Scholar
• L. Mei and X. Wu, Symplectic exponential Runge–Kutta methods for solving nonlinear Hamiltonian systems, J. Comput. Phys. 338 (2017), pp. 567–584.
   Web of Science ®Google Scholar
• X. Shen and M. Leok, Geometric exponential integrators, J. Comput. Phys. 382 (2019), pp. 27–42.
   Web of Science ®Google Scholar
• L.N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.
   Google Scholar
• B. Wang and X. Wu, Exponential collocation methods for conservative or dissipative systems, J. Comput. Appl. Math. 360 (2019), pp. 99–116.
   Web of Science ®Google Scholar
• B. Wang and X. Wu, Volume-preserving exponential integrators and their applications, J. Comput. Phys. 396 (2019), pp. 867–887.
   Web of Science ®Google Scholar
• X. Wu and B. Wang, Geometric Integrators for Differential Equations with Highly Oscillatory Solutions, Springer Nature Singapore Pte Ltd, 2021.
   Google Scholar
• B. Wang and X. Zhao, Error estimates of some splitting schemes for charged-particle dynamics under strong magnetic field, SIAM J. Numer. Anal. 59 (2021), pp. 2075–2105.
   Web of Science ®Google Scholar