Krylov subspace approximations for the exponential Euler method: error estimates and the harmonic Ritz approximant


  • Elliot Joseph Carr
  • Ian Turner
  • Milos Ilic



Krylov subspace methods, Matrix function approximation, Exponential integrators


We study Krylov subspace methods for approximating the matrix-function vector product $\varphi(tA)b$ where $\varphi(z) = [\exp(z)-1]/z$. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where $A$~is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to $\varphi(tA)b$, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error. References
  • E. J. Carr, T. J. Moroney and I. W. Turner. Efficient simulation of unsaturated flow using exponential time integration. Appl. Math. Comput., 217(14):6587--6596, 2011. doi:10.1016/j.amc.2011.01.041
  • E. J. Carr, I. W. Turner and P. Perre. A Jacobian-free exponential integrator for simulating transport in heterogeneous porous media: application to the drying of softwood, submitted for publication.
  • E. J. Carr, I. W. Turner and P. Perre. A new control-volume finite-element scheme for heterogeneous porous media:application to the drying of softwood. Chem. Eng. Technol., 34(7):1143--1150, 2011. doi:10.1002/ceat.201100060
  • E. Celledoni and I. Moret. A Krylov projection method for systems of ODEs. Appl. Numer. Math., 24(2--3):365--378, 1997. doi:10.1016/S0168-9274(97)00033-0
  • N. J. Higham. Functions of matrices: theory and computation. SIAM, Philadelphia, PA, USA, 2008.
  • M. Hochbruck, M. E. Hochstenbach. Subspace extraction for matrix functions, submitted for publication. hochsten/pdf/funext.pdf
  • M. Hochbruck, C. Lubich and H. Selhofer. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19(5):1552--1574, 1998. doi:10.1137/S1064827595295337
  • B. V. Minchev and W. M. Wright. A review of exponential integrators for first order semi-linear problems. Numerics No. 2/05, Norwegian University of Science and Technology, Trondheim, Norway, 2005.
  • P. Perre and I. Turner. A heterogeneous wood drying computational model that accounts for material property variation across growth rings. Chem. Eng. J., 86(1--2):117--131, 2002. doi:10.1016/S1385-8947(01)00270-4





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