Krylov subspace methods on supercomputers
SIAM Journal on Scientific and Statistical Computing
Fourth-order symplectic integration
Physica D
A symplectic integration algorithm for separable Hamiltonian functions
Journal of Computational Physics
Analysis of some Krylov subspace approximations to the matrix exponential operator
SIAM Journal on Numerical Analysis
Efficient solution of parabolic equations by Krylov approximation methods
SIAM Journal on Scientific and Statistical Computing
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
Nodal high-order methods on unstructured grids
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations
Journal of Computational Physics
Journal of Computational Physics
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows
Journal of Computational Physics
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We investigate efficient algorithms and a practical implementation of an explicit-type high-order timestepping method based on Krylov subspace approximations, for possible application to large-scale engineering problems in electromagnetics. We consider a semi-discrete form of the Maxwell's equations resulting from a high-order spectral-element discontinuous Galerkin discretization in space whose solution can be expressed analytically by a large matrix exponential of dimension $$\kappa \times \kappa $$ 驴 脳 驴 . We project the matrix exponential into a small Krylov subspace by the Arnoldi process based on the modified Gram---Schmidt algorithm and perform a matrix exponential operation with a much smaller matrix of dimension $$m\times m$$ m 脳 m ( $$m\ll \kappa $$ m 驴 驴 ). For computing the matrix exponential, we obtain eigenvalues of the $$m\times m$$ m 脳 m matrix using available library packages and compute an ordinary exponential function for the eigenvalues. The scheme involves mainly matrix-vector multiplications, and its convergence rate is generally $$O(\Delta t^{m-1})$$ O ( Δ t m - 1 ) in time so that it allows taking a larger timestep size as $$m$$ m increases. We demonstrate CPU time reduction compared with results from the five-stage fourth-order Runge---Kutta method for a certain accuracy. We also demonstrate error behaviors for long-time simulations. Case studies are also presented, showing loss of orthogonality that can be recovered by adding a low-cost reorthogonalization technique.