TITLE

# Fourier transform on sparse grids: Code design and the time dependent Schrï¿½dinger equation

AUTHOR(S)
PUB. DATE
May 2007
SOURCE
Computing;May2007, Vol. 80 Issue 1, p1
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
The pseudo-spectral method together with a Strang-splitting are well suited for the discretization of the time-dependent Schrï¿½dinger equation with smooth potential. The curse of dimensionality limits this approach to low dimensions, if we stick to full grids. Theoretically, sparse grids allow accurate computations in (moderately) higher dimensions, provided that we supply an efficient Fourier transform. Motivated by this application, the design of the Fourier transform on sparse grids in multiple dimensions is described in detail. The focus of this presentation is on issues of flexible implementation and numerical studies of the convergence.
ACCESSION #
25064519

## Related Articles

• Time decay for solutions of Schrï¿½dinger equations with rough and time-dependent potentials. Rodnianski, Igor; Schlag, Wilhelm // Inventiones Mathematicae;Mar2004, Vol. 155 Issue 3, p451

In this paper we establish dispersive estimates for solutions to the linear Schrï¿½dinger equation in three dimensions (0.1) 1/i ?t? - ?? + V? = 0, ?(s) = F where V(t, x) is a time-dependent potential that satisfies the conditions sup...V(t,ï¿½)...LZ(Rï¿½) + ...?Rï¿½?-88...dtau;dy <...

• On dynamics of quantum states generated by the Cauchy problem for the SchrÃ¶dinger equation with degeneration on the half-line. Sakbaev, V. // Journal of Mathematical Sciences;Jun2008, Vol. 151 Issue 1, p2741

The paper considers the Cauchy problem for the SchrÃ¶dinger equation with operator degenerate on the semiaxis and the family of regularized Cauchy problems with uniformly elliptic operators whose solutions approximate the solution of the degenerate problem. The author studies the strong and...

• Schrï¿½dinger Differential Equation and Wave Pockets for Elementary Particles in the Minkowski Spaces. Majkic, Zoran // Differential Equations & Control Processes;2011, Issue 3, p46

In this paper we presented the partial differential equations for the wave-pockets in the Minkowski 4-dimensional spaces, and their relationship with the Schrï¿½dinger equation for the elementary particles[7]. In this paper we show that the Schrï¿½dinger equation does not describe the...

• Standing Ring Blow up Solutions to the N-Dimensional Quintic Nonlinear Schrï¿½dinger Equation. Rapha�l, Pierre; Szeftel, J�r�mie // Communications in Mathematical Physics;Sep2009, Vol. 290 Issue 3, p973

We consider the quintic nonlinear Schrï¿½dinger equation $${i\partial_tu=-\Delta u-|u|^{4}u}$$ in dimension N = 3. This problem is energy critical in dimension N = 3 and energy super critical for N = 4. We prove the existence of a radially symmetric blow up mechanism with L2 concentration...

• The Boussinesq equation and Miura-type transformations. Pavlov, M. // Journal of Mathematical Sciences;Aug2006, Vol. 136 Issue 6, p4478

Several Miura-type transformations for the Boussinesq equation are found and the corresponding integrable systems are constructed.

• Self-similar solutions of Schrï¿½dinger flows. Ding, Weiyue; Tang, Hongyan; Zeng, Chongchun // Calculus of Variations & Partial Differential Equations;Feb2009, Vol. 34 Issue 2, p267

In this note, we showed the existence of equivariant self-similar solutions with finite local energy for the Schrï¿½dinger flow from $${\mathbb{C}^n}$$ into $${\mathbb{C}P^n}$$ ( n = 2).

• 1-Soliton solution of 1+2 dimensional nonlinear SchrÃ¶dinger's equation in parabolic law media. Biswas, Anjan // Nonlinear Studies;2009, Vol. 16 Issue 1, p49

This paper obtains an exact l-soliton solution of the nonlinear SchrÃ¶dinger's equation in 1+2 dimensions with parabolic law nonlinearity. The solitary wave ansatz is used to obtain this solution. In addition, a few of the conserved quantities are also computed using this exact solution.

• Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation II: The Low Density Regime. Benedetto, D.; Castella, F.; Esposito, R.; Pulvirenti, M. // Journal of Statistical Physics;Jul2006, Vol. 124 Issue 2-4, p951

In this paper we analyse the asymptotic dynamics of a system of N identical quantum particles in a low-density regime. Our approach follows the strategy introduced by the authors in a previous work,(2) to treat the simpler weak coupling regime. The time evolution of the Wigner transform of the...

• On the Pringsheim convergence of double series. Limaye, Balmohan V.; Zeltser, Maria // Proceedings of the Estonian Academy of Sciences;2009, Vol. 58 Issue 2, p108

Several aspects of the convergence of a double series in the sense of Pringsheim are considered in analogy with some well-known results for single series. They include various tests for absolute convergence and also criteria for convergence of the Cauchy product. Some errors in the works of...

Share