On the representation of operators in bases of compactly supported wavelets
SIAM Journal on Numerical Analysis
On the wavelet based differentiation matrix
Journal of Scientific Computing
The Differentiation Matrix for Daubechies-Based Wavelets on an Interval
SIAM Journal on Scientific Computing
Adaptive multiresolution collocation methods for initial boundary value problems of nonlinear PDEs
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases
Journal of Computational Physics
On the Gibbs Phenomenon and Its Resolution
SIAM Review
An adaptive spline wavelet ADI (SW-ADI) method for two-dimensional reaction-diffusion equations
Journal of Computational Physics
A wavelet optimized adaptive multi-domain method
Journal of Computational Physics
A Wavelet-Optimized, Very High Order Adaptive Grid and Order Numerical Method
SIAM Journal on Scientific Computing
Spectral methods for hyperbolic problems
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
Simulating 2D Waves Propagation in Elastic Solid Media Using Wavelet Based Adaptive Method
Journal of Scientific Computing
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We discuss a scheme for the numerical solution of one-dimensional initial value problems exhibiting strongly localized solutions or finite-time singularities. To accurately and efficiently model such phenomena we present a full space-time adaptive scheme, based on a variable order spatial finite-difference scheme and a 4th order temporal integration with adaptively chosen time step. A wavelet analysis is utilized at regular intervals to adaptively select the order and the grid in accordance with the local behavior of the solution. Through several examples, taken from gasdynamics and nonlinear optics, we illustrate the performance of the scheme, the use of which results in several orders of magnitude reduction in the required degrees of freedom to solve a problem to a particular fidelity.