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A Third-Order Low-Regularity Trigonometric Integrator for the Semilinear Klein-Gordon Equation


Core Concepts
A novel third-order low-regularity trigonometric integrator is constructed and analyzed for the semilinear Klein-Gordon equation in 1D, 2D, and 3D. The integrator achieves third-order accuracy in the energy space under a weak regularity requirement on the initial data.
Abstract
The paper proposes and analyzes a novel third-order low-regularity trigonometric integrator for the semilinear Klein-Gordon equation in d-dimensional space with d = 1, 2, 3. The key ideas are: Embedding the structure of the semilinear Klein-Gordon equation in the formulation. Applying the technique of twisted function to the trigonometric integrals appearing in the Duhamel's formula. The proposed integrator is shown to have third-order convergence in the H1 × L2 energy space under the weaker regularity condition (u0, v0) ∈ H2 × H1, compared to the classical third-order methods which require (u0, v0) ∈ H3 × H2. The paper also presents a fully-discrete scheme and studies the error bounds from both spatial discretization and semi-discretization in time for the semilinear Klein-Gordon equation with nonsmooth initial data.
Stats
The semilinear Klein-Gordon equation is globally well-posed for initial data (u0, v0) ∈ Hγ(Td) × Hγ-1(Td) for γ ≥ 1 in high dimensions. Classical time discretization methods generally require strong regularity assumptions, e.g., (u0, v0) ∈ Hγ+m-1(Td) × Hγ+m-2(Td) to achieve mth-order approximation in Hγ(Td) × Hγ-1(Td). The proposed third-order low-regularity integrator only requires the initial data (u0, v0) to be in H2(Td) × H1(Td).
Quotes
"To overcome this barrier, much attention has been paid to the equations with nonsmooth initial data recently." "It is an interesting question whether higher-order algorithm can achieve low regularity property. The answer is positive but the construction of higher-order algorithms is very challenging."

Deeper Inquiries

How can the proposed third-order low-regularity integrator be extended to other nonlinear wave equations beyond the Klein-Gordon equation

The proposed third-order low-regularity integrator for the Klein-Gordon equation can be extended to other nonlinear wave equations by adapting the same principles and techniques used in its construction. The key idea is to carefully analyze the structure of the specific wave equation, utilize Duhamel's formula, and apply twisted functions to handle the trigonometric integrals. By understanding the underlying dynamics and regularity requirements of the new wave equation, one can modify the integrator to suit the specific characteristics of the equation. This may involve adjusting the interpolation operators, co-efficient functions, and error estimates to ensure convergence and accuracy for the new equation.

What are the potential limitations or drawbacks of the low-regularity approach compared to classical high-order methods when the initial data is sufficiently smooth

While low-regularity integrators offer advantages in handling nonsmooth initial data, they may have limitations compared to classical high-order methods when dealing with sufficiently smooth initial conditions. Some potential drawbacks include: Accuracy: High-order methods typically provide higher accuracy and precision in approximating the solution compared to low-regularity integrators. This can be crucial for applications requiring very precise results. Computational Cost: Low-regularity integrators may require more computational resources or time compared to high-order methods, especially for problems with smooth solutions where high accuracy is essential. Convergence Rates: High-order methods often exhibit faster convergence rates, leading to quicker and more efficient computations, particularly for problems with smooth solutions.

What are the connections between the low-regularity integrators developed for the Klein-Gordon equation and other dispersive PDEs, and can the techniques be unified into a more general framework

The connections between low-regularity integrators developed for the Klein-Gordon equation and other dispersive PDEs lie in the underlying principles and techniques used in their construction. While the specific details may vary based on the characteristics of each equation, the general framework of utilizing Duhamel's formula, twisted functions, and trigonometric integrals can be applied to a wide range of dispersive PDEs. By understanding the regularity requirements, the nonlinearities, and the specific dynamics of each equation, the techniques developed for the Klein-Gordon equation can be adapted and unified into a more general framework for solving various dispersive PDEs. This unified approach can provide a systematic and efficient way to develop low-regularity integrators for a broad class of wave equations.
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