Scientific computing: an introduction with parallel computing
Scientific computing: an introduction with parallel computing
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
Spectral methods in MatLab
Exponential time differencing for stiff systems
Journal of Computational Physics
Frequency evaluation in exponential fitting multistep algorithms for ODEs
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
On the stability of exponential fitting BDF algorithms
Journal of Computational and Applied Mathematics - Special issue: Selected papers of the international conference on computational methods in sciences and engineering (ICCMSE-2003)
Exponential fitting BDF algorithms: explicit and implicit 0-stable methods
Journal of Computational and Applied Mathematics - Special issue on computational and mathematical methods in science and engineering (CMMSE-2004)
On smoothing of the Crank-Nicolson scheme and higher order schemes for pricing barrier options
Journal of Computational and Applied Mathematics
Spectral collocation solution of a generalized Hirota-Satsuma coupled KdV equation
International Journal of Computer Mathematics
Numerical solutions of some nonlinear evolution equations by Chebyshev spectral collocation methods
International Journal of Computer Mathematics
Stabilized explicit Runge-Kutta methods for multi-asset American options
Computers & Mathematics with Applications
| Hi-index | 7.29 |
Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002) 430-455] developed a class of Exponential Time Differencing Runge-Kutta schemes (ETDRK) for nonlinear parabolic equations; Kassam and Trefethen [A.K. Kassam, Ll. N. Trefethen, Fourth-order time stepping for stiff pdes, SIAM J. Sci. Comput. 26 (2005) 1214-1233] have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth-order (ETDRK4) scheme. They use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials, but this approach creates new difficulties in choosing and evaluating the contour for larger problems. Neither treatment addresses problems with nonsmooth data, where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully. Such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications, including computational chemistry and financial engineering. We introduce a new version of the fourth-order Cox-Matthews, Kassam-Trefethen ETDRK4 scheme designed to eliminate the remaining computational difficulties. This new scheme utilizes an exponential time differencing Runge-Kutta ETDRK scheme using a diagonal Pade approximation of matrix exponential functions, while to deal with the problem of nonsmooth data we use several steps of an ETDRK scheme using a sub-diagonal Pade formula. The new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices, having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel. In this approach it is only required that several backward Euler linear problems be solved, in serial or parallel. Numerical experiments are described to support the new scheme.