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A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions

Weizhu Bao, Caoyi Liu

TL;DR

The NKGE in the nonrelativistic regime exhibits highly oscillatory time behavior as $\varepsilon\to0$, challenging uniform accuracy. The paper introduces MTI-FP, a frequency-based multiscale decomposition combined with an exponential wave integrator and a Fourier pseudospectral discretization, achieving uniform-in-$\varepsilon$ first-order time accuracy and spectral spatial accuracy. A multiscale time interpolation extends the method to provide uniformly accurate solutions for all $t\ge0$, demonstrating a super-resolution property in time. Numerical experiments validate the theoretical error bounds, showcase convergence to the limiting NLSW and NLSE models, and highlight the method’s efficiency in higher dimensions.

Abstract

We propose a new and simplified multiscale time integrator Fourier pseudospectral (MTI-FP) method for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter epsilon in (0,1] inversely proportional to the speed of light, and establish its uniform first-order accuracy in time in the nonrelativistic regime, i.e. 0 < epsilon << 1. In this regime, the solution of the NKGE is highly oscillatory in time with O(epsilon^2)-wavelength, which brings significant difficulties in designing uniformly accurate numerical methods. The MTI-FP is based on (i) a multiscale decomposition by frequency of the NKGE in each time interval with simplified transmission conditions, and (ii) an exponential wave integrator for temporal discretization and a Fourier pseudospectral method for spatial discretization. By adapting the energy method and the mathematical induction, we obtain two error bounds in H1-norm at O(h^{m0}+tau^2/epsilon^2) and O(h^{m0}+tau+epsilon^2) with mesh size h, time step tau and m0 an integer dependent on the regularity of the solution, which immediately implies a uniformly accurate error bound O(h^{m0}+tau) with respect to epsilon in (0,1]. In addition, by adopting a linear interpolation of the micro variables with the multiscale decomposition in each time interval, we obtain a uniformly accurate numerical solution for any time t larger than zero. Thus the proposed MTI-FP method has a super resolution property in time in terms of the Shannon sampling theory, i.e. accurate numerical solutions can be obtained even when the time step is much bigger than the O(epsilon^2)-wavelength. Extensive numerical results are reported to confirm our error bounds and demonstrate their super resolution in time. Finally the proposed MTI-FP method is applied to study numerically convergence rates of the NKGE to its different limiting models in the nonrelativistic regime.

A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions

TL;DR

The NKGE in the nonrelativistic regime exhibits highly oscillatory time behavior as , challenging uniform accuracy. The paper introduces MTI-FP, a frequency-based multiscale decomposition combined with an exponential wave integrator and a Fourier pseudospectral discretization, achieving uniform-in- first-order time accuracy and spectral spatial accuracy. A multiscale time interpolation extends the method to provide uniformly accurate solutions for all , demonstrating a super-resolution property in time. Numerical experiments validate the theoretical error bounds, showcase convergence to the limiting NLSW and NLSE models, and highlight the method’s efficiency in higher dimensions.

Abstract

We propose a new and simplified multiscale time integrator Fourier pseudospectral (MTI-FP) method for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter epsilon in (0,1] inversely proportional to the speed of light, and establish its uniform first-order accuracy in time in the nonrelativistic regime, i.e. 0 < epsilon << 1. In this regime, the solution of the NKGE is highly oscillatory in time with O(epsilon^2)-wavelength, which brings significant difficulties in designing uniformly accurate numerical methods. The MTI-FP is based on (i) a multiscale decomposition by frequency of the NKGE in each time interval with simplified transmission conditions, and (ii) an exponential wave integrator for temporal discretization and a Fourier pseudospectral method for spatial discretization. By adapting the energy method and the mathematical induction, we obtain two error bounds in H1-norm at O(h^{m0}+tau^2/epsilon^2) and O(h^{m0}+tau+epsilon^2) with mesh size h, time step tau and m0 an integer dependent on the regularity of the solution, which immediately implies a uniformly accurate error bound O(h^{m0}+tau) with respect to epsilon in (0,1]. In addition, by adopting a linear interpolation of the micro variables with the multiscale decomposition in each time interval, we obtain a uniformly accurate numerical solution for any time t larger than zero. Thus the proposed MTI-FP method has a super resolution property in time in terms of the Shannon sampling theory, i.e. accurate numerical solutions can be obtained even when the time step is much bigger than the O(epsilon^2)-wavelength. Extensive numerical results are reported to confirm our error bounds and demonstrate their super resolution in time. Finally the proposed MTI-FP method is applied to study numerically convergence rates of the NKGE to its different limiting models in the nonrelativistic regime.
Paper Structure (18 sections, 7 theorems, 147 equations, 5 figures, 2 tables)

This paper contains 18 sections, 7 theorems, 147 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Multiscale decompositions by amplitude and/or frequency
  3. A uniformly accurate MTI-FP method
  4. A multiscale decomposition via simplified transmission conditions
  5. An exponential wave integrator Fourier pseudospectral full discretization
  6. A uniformly accurate error bound
  7. The main result
  8. Estimates for the micro (or local) variables
  9. Error functions and the corresponding error equations
  10. Energy estimates for error functions
  11. Proof for Theorem \ref{['tm4.1']} by the energy method
  12. A multiscale interpolation in time and its uniformly accurate error bounds
  13. The multiscale interpolation in time
  14. A uniformly accurate error bound
  15. Numerical results
  16. ...and 3 more sections

Key Result

Theorem 1

Under the assumption (A), there exist two constants $0 < h_0 < 1$ and $0 < \tau_0 < 1$ sufficiently small and independent of $\varepsilon$ such that for any $0 < \varepsilon \le 1$, when $0 < h \le h_0$ and $0 < \tau \le \tau_0$, we have Thus, by taking the minimum of the two error estimates in thm1err1 and then taking the maximum for $\varepsilon \in (0,1]$, we obtain an error estimate uniformly

Figures (5)

  • Figure 1: Illustration of the multiscale interpolation \ref{['MTIint2']} with $\tau=0.5$, $v_{0}^{0,0}=0.5$, $v_{0}^{0,1}=1.0$ and $r_{0}^{0,1}=\varepsilon^2$ for different $\varepsilon$: (a) $\varepsilon=0.5$, (b) $\varepsilon=0.25$, and (c) $\varepsilon=0.125$.
  • Figure 2: Plots of the multiscale interpolation error $|u(0,t)-U_{N/2}(t)|$ with $U_{N/2}(t)$ obtained via the multiscale interpolation \ref{['MTIint1']}-\ref{['MTIint2']} for different $\varepsilon$ and $\tau$ with $N=2^{10}$: (a) $\varepsilon=0.5$, (b) $\varepsilon=0.05$, and (c) $\varepsilon=0.005$.
  • Figure 3: Dynamics of the solution $u$ of the NKGE \ref{['1.1']} in two dimensions for $\varepsilon= 1$ (left column) and $\varepsilon = 0.01$ (right column).
  • Figure 4: Convergence of the NKGE \ref{['NKGE1d']} to its limiting model NLSW \ref{['NLSW1']} with different $\gamma$ in 1D: (a) $\gamma(x)\equiv 0$, (b) $\gamma(x)$ is taken as the well-prepared initial data \ref{['2.12']}, and (c) $\gamma(x) =\frac{3}{2} \lambda i |v_0(x)|^2 v_0(x)$ with $v_0(x)$ given in \ref{['initv23']}.
  • Figure 5: Convergence of the NLSW \ref{['NLSW1']} to its limiting model NLSE \ref{['NLSE1']} with different $\gamma$ in 1D: (a) $\gamma(x)\equiv 0$, (b) $\gamma(x)$ is taken as the well-prepared initial data \ref{['2.12']}, and (c) $\gamma(x) =\frac{3}{2} \lambda i |v_0(x)|^2 v_0(x)$ with $v_0(x)$ given in \ref{['initv23']}.

Theorems & Definitions (19)

  • Remark 1
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 9 more