46 research outputs found
A symmetric low-regularity integrator for the nonlinear Schr\"odinger equation
We introduce and analyze a symmetric low-regularity scheme for the nonlinear
Schr\"odinger (NLS) equation beyond classical Fourier-based techniques. We show
fractional convergence of the scheme in -norm, from first up to second
order, both on the torus and on a smooth bounded domain , , equipped with homogeneous Dirichlet boundary
condition. The new scheme allows for a symmetric approximation to the NLS
equation in a more general setting than classical splitting, exponential
integrators, and low-regularity schemes (i.e. under lower regularity
assumptions, on more general domains, and with fractional rates). We motivate
and illustrate our findings through numerical experiments, where we witness
better structure preserving properties and an improved error-constant in
low-regularity regimes
Error Analysis of Exponential Integrators for Nonlinear Wave-Type Equations
This thesis is concerned with the time integration of certain classes of nonlinear evolution equations in Hilbert spaces by exponential integrators. We aim to prove error bounds which can be established by including only quantities given by a wellposedness result. In the first part, we consider semilinear wave equations and introduce a class of first- and second-order exponential schemes. A standard error analysis is not possible due to the lack of regularity. We have to employ appropriate filter functions as well as the integration by parts and summation by parts formulas in order to obtain optimal error bounds. In the second part, we propose two exponential integrators of first and second order applied to a class of quasilinear wave-type equations. By a detailed investigation of the differentiability of the right-hand side we derive error bounds in different norms. In the framework we can treat quasilinear Maxwell’s equations in full space and on a smooth domain as well as a class of quasilinear wave equations. In both parts, we include numerical examples to confirm our theoretical findings
Exponential integrators: tensor structured problems and applications
The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed
On numerical methods for the semi-nonrelativistic system of the nonlinear Dirac equation
Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of , where is inversely proportional to the speed of light. It was shown in [7], however, that such solutions can be approximated up to an error of by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the exponential explicit midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy by a factor of six
An explicit and symmetric exponential wave integrator for the nonlinear Schr\"{o}dinger equation with low regularity potential and nonlinearity
We propose and analyze a novel symmetric exponential wave integrator (sEWI)
for the nonlinear Schr\"odinger equation (NLSE) with low regularity potential
and typical power-type nonlinearity of the form ,
where is the density with the wave function and is the exponent of the nonlinearity. The sEWI is explicit and
stable under a time step size restriction independent of the mesh size. We
rigorously establish error estimates of the sEWI under various regularity
assumptions on potential and nonlinearity. For "good" potential and
nonlinearity (-potential and ), we establish an optimal
second-order error bound in -norm. For low regularity potential and
nonlinearity (-potential and ), we obtain a first-order
-norm error bound accompanied with a uniform -norm bound of the
numerical solution. Moreover, adopting a new technique of \textit{regularity
compensation oscillation} (RCO) to analyze error cancellation, for some
non-resonant time steps, the optimal second-order -norm error bound is
proved under a weaker assumption on the nonlinearity: . For
all the cases, we also present corresponding fractional order error bounds in
-norm, which is the natural norm in terms of energy. Extensive numerical
results are reported to confirm our error estimates and to demonstrate the
superiority of the sEWI, including much weaker regularity requirements on
potential and nonlinearity, and excellent long-time behavior with
near-conservation of mass and energy.Comment: 35 pages, 10 figure
Lowregularity exponential-type integrators for semilinear Schrödinger equations
We introduce low regularity exponential-type integrators for nonlinear Schrödinger equations for which first-order convergence only requires the boundedness of one additional derivative of the solution. More precisely, we will prove first-order convergence in H for solutions in H (r > d/2) of the derived schemes. This allows us lower regularity assumptions on the data than for instance required for classical splitting or exponential integration schemes. For one dimensional quadratic Schrödinger equations we can even prove first-order convergence without any loss of regularity. Numerical experiments underline the favorable error behavior of the newly introduced exponential-type integrators for low regularity solutions compared to classical splitting and exponential integration schemes