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
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
A new class of time discretization schemes for the solution of nonlinear PDEs
Journal of Computational Physics
Exponential time differencing for stiff systems
Journal of Computational Physics
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
Generalized integrating factor methods for stiff PDEs
Journal of Computational Physics
Efficient semi-implicit schemes for stiff systems
Journal of Computational Physics
Compact integration factor methods in high spatial dimensions
Journal of Computational Physics
Fokker–Planck approximation of the master equation in molecular biology
Computing and Visualization in Science
Removing the stiffness from interfacial flows with surface tension
Journal of Computational Physics
Compact integration factor methods for complex domains and adaptive mesh refinement
Journal of Computational Physics
Journal of Computational Physics
Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems
SIAM Journal on Scientific Computing
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High order spatial derivatives and stiff reactions often introduce severe temporal stability constraints on the time step in numerical methods. Implicit integration method (IIF) method, which treats diffusion exactly and reaction implicitly, provides excellent stability properties with good efficiency by decoupling the treatment of reactions and diffusions. One major challenge for IIF is storage and calculation of the potential dense exponential matrices of the sparse discretization matrices resulted from the linear differential operators. Motivated by a compact representation for IIF (cIIF) for Laplacian operators in two and three dimensions, we introduce an array-representation technique for efficient handling of exponential matrices from a general linear differential operator that may include cross-derivatives and non-constant diffusion coefficients. In this approach, exponentials are only needed for matrices of small size that depend only on the order of derivatives and number of discretization points, independent of the size of spatial dimensions. This method is particularly advantageous for high-dimensional systems, and it can be easily incorporated with IIF to preserve the excellent stability of IIF. Implementation and direct simulations of the array-representation compact IIF (AcIIF) on systems, such as Fokker-Planck equations in three and four dimensions and chemical master equations, in addition to reaction-diffusion equations, show efficiency, accuracy, and robustness of the new method. Such array-presentation based on methods may have broad applications for simulating other complex systems involving high-dimensional data.