Fourth‐order compact and energy conservative scheme for solving nonlinear Klein‐Gordon equation

We present the fourth-order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE), while the nonlinearity strength is characterized by p with a constant p ∈ N + and a dimensionless parameter ∈ (0, 1]. Based on analytical results of the lifespan of the solution, rigorous error bounds of the 4cFD methods are carried out up to the time at O(−p). We pay particular attention to how error bounds depend explicitly on the mesh size h and time step as well as the small parameter ∈ (0, 1], which indicate that, in order to obtain 'correct' numerical solutions up to the time at O(−p), the-scalability (or meshing strategy requirement) of the 4cFD methods should be taken as: h = O(p/4) and = O(p/2). It has better spatial resolution capacity than the classical second order central difference methods. By a rescaling in time, it is equivalent to an oscillatory NKGE whose solution propagates waves with wavelength at O(1) in space and O(p) in time. It is straightforward to get the error bounds of the oscillatory NKGE in the fixed time. Finally, numerical results are provided to confirm our theoretical analysis.

false(δ_{x}^{+} δ_{x}^{-} u, v false) = - false(δ_{x}^{+} u, δ_{x}^{+} v false) .

Lemma The operators

A

and

{scriptA}^{- 1}

are commutative with

δ_{x}^{+}

and

δ_{x}^{-}

, that is, for any grid function u ∈ X M ,

\begin{array}{l} δ_{x}^{+} scriptA u = scriptA δ_{x}^{+} u, δ_{x}^{-} scriptA u = scriptA δ_{x}^{-} u, \\ δ_{x}^{+} A^{- 1} u = A^{- 1} δ_{x}^{+} u, δ_{x}^{-} A^{- 1} u = A^{- 1} δ_{x}^{-} u . \end{array}

Lemma The discrete norms ‖ ⋅ ‖ * and

‖ \cdot ‖_{l^{2}}

are equivalent. In fact, for any grid function u ∈ X M , it holds

‖ u ‖_{l^{2}} \leq ‖ u ‖_{*} \leq \frac{\sqrt{6}}{2} ‖ u ‖_{l^{2}} .

…”

Section: Cfd Methods and Their Analysismentioning

confidence: 95%

“…x [31,32,[34][35][36]. In this paper, we consider the following fourth-order compact finite difference (4cFD) methods:…”

Section: Cfd Methodsmentioning

confidence: 99%

“…A fourth‐order approximation is implemented by replacing the central difference operator

δ_{x}^{2}

with

(1 - \frac{h^{2}}{12} δ_{x}^{2}) δ_{x}^{2}

, which requires a five‐point stencil. In order to obtain a compact three‐point stencil,

(1 - \frac{h^{2}}{12} δ_{x}^{2}) δ_{x}^{2}

is approximated by

{((), 1 + \frac{h^{2}}{12} δ_{x}^{2})}^{- 1} δ_{x}^{2}

[31, 32, 34–36].…”

Section: Cfd Methods and Their Analysismentioning

confidence: 99%

“…The following lemmas will be used in our error estimates. The proof proceeds in the analogous lines as in [34,36] and we just show the proof of Lemma 2.3 in detail here for brevity.…”

Section: Some Useful Lemmasmentioning

confidence: 97%

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Long time error analysis of the fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation with weak nonlinearity

Feng

2020

“…The spatial fourth-order compact finite difference operator h is defined as 6) where I denotes identical operator. From [18,23], one can see that…”

Section: Numerical Schemesmentioning

confidence: 99%

Optimal error estimates of fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation in the nonrelativistic regime

Zhang

Wang

2020

Two fourth-order compact finite difference schemes including a Crank-Nicolson one and a semi-implicit one are derived for solving the nonlinear Klein-Gordon equations in the nonrelativistic regime. The optimal error estimates and the strategy in choosing time step are rigorously analyzed, and the energy conservation in the discrete sense is also studied. Under proper assumption on the analytical solutions, the errors of the two schemes both are proved to be of O (h 4 + 2 6) with mesh size h and time-step τ. Numerical simulations are provided to confirm the theoretical analysis.

Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Feng

Ying

2022

We establish the error bounds of fourth-order compact finite difference (4cFD) methods for the Dirac equation in the massless and nonrelativistic regime, which involves a small dimensionless parameter 0 < 𝜀 ≤ 1 inversely proportional to the speed of light. In this regime, the solution propagates waves with wavelength O(𝜀) in time and O(1) in space, as well as with the wave speed O(1∕𝜀) rapid outgoing waves. We adapt the conservative and semi-implicit 4cFD methods to discretize the Dirac equation and rigorously carry out their error bounds depending explicitly on the mesh size h, time step 𝜏 and the small parameter 𝜀. Based on the error bounds, the ε-scalability of the 4cFD, which not only improves the spatial resolution capacity but also has superior accuracy than classical second-order finite difference methods. Furthermore, physical observables including the total density and current density have the same conclusions. Numerical results are provided to validate the error bounds and the dynamics of the Dirac equation with different potentials in two dimensions is presented.