แนวคิดหลัก
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.
บทคัดย่อ
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.
สถิติ
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).
คำพูด
"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."