论文标题
半连接klein-gordon方程的二阶低规范校正分裂的谎言校正
A second-order low-regularity correction of Lie splitting for the semilinear Klein--Gordon equation
论文作者
论文摘要
通过分析近似解决方案时的一致性错误，研究了$ d $维空间中半级klein方程的数值近似值，并具有$ d = 1,2,3 $。通过在半连续性klein方程中发现并利用新的取消结构，构建了谎言分裂方法的低规范性校正，在规则性条件$（u，\ partial_tu）在l^\ infty（u，partial_tu）\ in l^frac; h^{\ frac {d} {4}}）$，其中$ d = 1,2,3 $表示空间的维度。在一个维度中，所提出的方法显示出在同一空间中解决方案的能量空间中任意接近$ \ frac53 $的收敛顺序，即在解决方案中不需要其他规律性。使用拟议的低规范时间稳定方案提出了针对完全离散的光谱法的严格误差估计。提供了数值示例来支持理论分析，并说明了所提出的方法在近似半线性klein-grodon方程的非平滑和平滑溶液时的性能。
The numerical approximation of the semilinear Klein--Gordon equation in the $d$-dimensional space, with $d=1,2,3$, is studied by analyzing the consistency errors in approximating the solution. By discovering and utilizing a new cancellation structure in the semilinear Klein--Gordon equation, a low-regularity correction of the Lie splitting method is constructed, which can have second-order convergence in the energy space under the regularity condition $(u,\partial_tu)\in L^\infty(0,T;H^{1+\frac{d}{4}}\times H^{\frac{d}{4}})$, where $d=1,2,3$ denotes the dimension of space. In one dimension, the proposed method is shown to have a convergence order arbitrarily close to $\frac53$ in the energy space for solutions in the same space, i.e. no additional regularity in the solution is required. Rigorous error estimates are presented for a fully discrete spectral method with the proposed low-regularity time-stepping scheme. Numerical examples are provided to support the theoretical analysis and to illustrate the performance of the proposed method in approximating both nonsmooth and smooth solutions of the semilinear Klein--Gordon equation.
