论文标题
低规律性集成商的一般框架
A general framework of low regularity integrators
论文作者
论文摘要
我们引入了一个新的通用框架，用于低规律性以低规律性的进化方程的近似，并为在较低的规律性假设下开发了一类新的方案，而不是经典方法所要求的。与以前的作品相反，我们的新框架允许统一的实际配方，而新方案的构建不依赖于任何基于傅立叶的扩展。这使我们首次克服对周期性边界条件的严重限制，嵌入相同的框架抛物线和分散方程中，并处理非多项式的非线性。特别是，由于我们的新形式主义不再需要问题的周期性，因此，不仅可以将新的时间离散化技术与光谱方法相结合，而且还可以与各种空间离散化相结合。我们将一般理论应用于各种混凝土PDE的时间离散化，例如非线性热方程，非线性schrödinger方程，复杂的Ginzburg-landau方程，半波和klein-- klein-gordon方程，以$ω\ subset \ subset \ subbb {r}^d $，$ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ q copt
We introduce a new general framework for the approximation of evolution equations at low regularity and develop a new class of schemes for a wide range of equations under lower regularity assumptions than classical methods require. In contrast to previous works, our new framework allows a unified practical formulation and the construction of the new schemes does not rely on any Fourier based expansions. This allows us for the first time to overcome the severe restriction to periodic boundary conditions, to embed in the same framework parabolic and dispersive equations and to handle nonlinearities that are not polynomial. In particular, as our new formalism does no longer require periodicity of the problem, one may couple the new time discretisation technique not only with spectral methods, but rather with various spatial discretisations. We apply our general theory to the time discretization of various concrete PDEs, such as the nonlinear heat equation, the nonlinear Schrödinger equation, the complex Ginzburg-Landau equation, the half wave and Klein--Gordon equations, set in $Ω\subset \mathbb{R}^d$, $d \leq 3$ with suitable boundary conditions.
