Energy-preserving exponential integrators of arbitrarily high order for conservative or dissipative systems with highly oscillatory solutions

AbstractThis paper is concerned with the efficient numerical treatment of 1D stationary Schrödinger equations in the semi-classical limit when including a turning point of first order. As such it is an extension of the paper [3], where turning points still had to be excluded. For the considered scattering problems we show that the wave function asymptotically blows up at the turning point as the scaled Planck constant $$\varepsilon \rightarrow 0$$ε→0, which is a key challenge for the analysis. Assuming that the given potential is linear or quadratic in a small neighborhood of the turning point, the problem is analytically solvable on that subinterval in terms of Airy or parabolic cylinder functions, respectively. Away from the turning point, the analytical solution is coupled to a numerical solution that is based on a WKB-marching method—using a coarse grid even for highly oscillatory solutions. We provide an error analysis for the hybrid analytic-numerical problem up to the turning point (where the solution is asymptotically unbounded) and illustrate it in numerical experiments: if the phase of the problem is explicitly computable, the hybrid scheme is asymptotically correct w.r.t. $$\varepsilon $$ε. If the phase is obtained with a quadrature rule of, e.g., order 4, then the spatial grid size has the limitation $$h=\mathcal{O}(\varepsilon ^{7/12})$$h=O(ε7/12) which is slightly worse than the $$h=\mathcal{O}(\varepsilon ^{1/2})$$h=O(ε1/2) restriction in the case without a turning point.

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Geometric Integrators for Differential Equations with Highly Oscillatory Solutions

10.1007/978-981-16-0147-7 ◽

2021 ◽

Author(s):

Xinyuan Wu ◽

Bin Wang

Keyword(s):

Differential Equations ◽

Oscillatory Solutions ◽

Geometric Integrators ◽

Highly Oscillatory

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Simulation of Inviscid Compressible Flows Using PDE Transform

Communications in Computational Physics ◽

10.4208/cicp.031113.160514a ◽

2014 ◽

Vol 16 (5) ◽

pp. 1201-1238

Author(s):

Langhua Hu ◽

Siyang Yang ◽

Guo-Wei Wei

Keyword(s):

Conservation Laws ◽

Conservation Law ◽

Hyperbolic Conservation Laws ◽

High Order ◽

Shock Capturing ◽

Propagation Time ◽

Oscillatory Solutions ◽

Hyperbolic Conservation ◽

Time Frequency ◽

The Impact

AbstractThe solution of systems of hyperbolic conservation laws remains an interesting and challenging task due to the diversity of physical origins and complexity of the physical situations. The present work introduces the use of the partial differential equation (PDE) transform, paired with the Fourier pseudospectral method (FPM), as a new approach for hyperbolic conservation law problems. The PDE transform, based on the scheme of adaptive high order evolution PDEs, has recently been applied to decompose signals, images, surfaces and data to various target functional mode functions such as trend, edge, texture, feature, trait, noise, etc. Like wavelet transform, the PDE transform has controllable time-frequency localization and perfect reconstruction. A fast PDE transform implemented by the fast Fourier Transform (FFT) is introduced to avoid stability constraint of integrating high order PDEs. The parameters of the PDE transform are adaptively computed to optimize the weighted total variation during the time integration of conservation law equations. A variety of standard benchmark problems of hyperbolic conservation laws is employed to systematically validate the performance of the present PDE transform based FPM. The impact of two PDE transform parameters, i.e., the highest order and the propagation time, is carefully studied to deliver the best effect of suppressing Gibbs’ oscillations. The PDE orders of 2-6 are used for hyperbolic conservation laws of low oscillatory solutions, while the PDE orders of 8-12 are often required for problems involving highly oscillatory solutions, such as shock-entropy wave interactions. The present results are compared with those in the literature. It is found that the present approach not only works well for problems that favor low order shock capturing schemes, but also exhibits superb behavior for problems that require the use of high order shock capturing methods.

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High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations

Computer Physics Communications ◽

10.1016/j.cpc.2017.07.016 ◽

2017 ◽

Vol 220 ◽

pp. 243-262 ◽

Cited By ~ 17

Author(s):

Sergio Blanes ◽

Fernando Casas ◽

Mechthild Thalhammer

Keyword(s):

Evolution Equations ◽

High Order ◽

Exponential Integrators ◽

Linear Evolution ◽

Linear Evolution Equations

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Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations

IMA Journal of Numerical Analysis ◽

10.1093/imanum/drz058 ◽

2020 ◽

Author(s):

Sergio Blanes ◽

Fernando Casas ◽

Cesáreo González ◽

Mechthild Thalhammer

Keyword(s):

Evolution Equations ◽

High Order ◽

Time Dependent ◽

Schrodinger Equations ◽

Local Error ◽

Operator Family ◽

Schrödinger Equations ◽

Unbounded Operators ◽

Exponential Integrators ◽

Variation Of Constants Formula

Abstract This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrödinger equations; a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix; this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge–Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved; this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker–Campbell–Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrödinger equations is related to an appropriate treatment of compositions of evolution operators; an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-of-constants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.

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