Matrix-free methods for stiff systems of ODE's
SIAM Journal on Numerical Analysis
Hybrid Krylov methods for nonlinear systems of equations
SIAM Journal on Scientific and Statistical Computing
Accelerated Inexact Newton Schemes for Large Systems of Nonlinear Equations
SIAM Journal on Scientific Computing
Design and Application of a Gradient-Weighted Moving Finite Element Code I: in One Dimension
SIAM Journal on Scientific Computing
Design and Application of a Gradient-Weighted Moving Finite Element Code II: in Two Dimensions
SIAM Journal on Scientific Computing
Multi-Adaptive Galerkin Methods for ODEs I
SIAM Journal on Scientific Computing
Multi-Adaptive Galerkin Methods for ODEs II: implementation and Applications
SIAM Journal on Scientific Computing
Software frameworks for the computational simulation of structural systems
Software frameworks for the computational simulation of structural systems
Anderson Acceleration for Fixed-Point Iterations
SIAM Journal on Numerical Analysis
A bidirectional coupling procedure applied to multiscale respiratory modeling
Journal of Computational Physics
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We report on some successes and problem areas in the Method of Lines from our work with moving node finite element methods. First, we report on our ''nonlinear Krylov accelerator'' for the modified Newton's method on the nonlinear equations of our stiff ODE solver. Since 1990 it has been robust, simple, cheap, and automatic on all our moving node computations. We publicize further trials with it here because it should be of great general usefulness to all those solving evolutionary equations. Second, we discuss the need for reliable automatic choice of spatially variable time steps. Third, we discuss the need for robust and efficient iterative solvers for the difficult linearized equations (Jx=b) of our stiff ODE solver. Here, the 1997 thesis of Zulu Xaba has made significant progress.