论文标题
Monge-Ampère方程的基于趋同的正交方法
A Convergent Quadrature Based Method For The Monge-Ampère Equation
论文作者
论文摘要
我们介绍了Monge-Ampère方程的整体表示，该方程导致基于数值正交的新有限差方法。最终的方案是单调的，并立即适合使用Dirichlet或最佳传输边界条件的Monge-ampère方程的现有融合证明。高阶正交方案的使用可以大大减少误差的组成部分，这取决于有限差模板的角度分辨率。反过来，这允许在模板宽度和正式截断误差方面进行显着改善。结果方案可以实现正式的准确性，该准确性是任意接近$ \ Mathcal {o}（H^2）$的正式准确性，这是二阶运算符单调近似值的最佳一致性顺序。我们提出了该方法的三种不同实现。前两个利用梯形规则在均匀的角度离散上利用光谱准确性，以便在最近的邻骨有限差模板上计算在大量的网格细节上。第三个使用高阶正交来产生超线性收敛，同时利用与其他单调方法相比，同时使用狭窄的模板。计算结果在二维中出现了各种规律性问题。
We introduce an integral representation of the Monge-Ampère equation, which leads to a new finite difference method based upon numerical quadrature. The resulting scheme is monotone and fits immediately into existing convergence proofs for the Monge-Ampère equation with either Dirichlet or optimal transport boundary conditions. The use of higher-order quadrature schemes allows for substantial reduction in the component of the error that depends on the angular resolution of the finite difference stencil. This, in turn, allows for significant improvements in both stencil width and formal truncation error. The resulting schemes can achieve a formal accuracy that is arbitrarily close to $\mathcal{O}(h^2)$, which is the optimal consistency order for monotone approximations of second order operators. We present three different implementations of this method. The first two exploit the spectral accuracy of the trapezoid rule on uniform angular discretizations to allow for computation on a nearest-neighbors finite difference stencil over a large range of grid refinements. The third uses higher-order quadrature to produce superlinear convergence while simultaneously utilizing narrower stencils than other monotone methods. Computational results are presented in two dimensions for problems of various regularity.
