守恒律高阶方法中作为非振荡限制器的精确总变分极小化器
Exact Total Variation Minimizers as Non-Oscillatory Limiters in High-Order Methods for Conservation Laws
AI总结：
该研究针对守恒律高阶方法的伪振荡问题，提出用微分包含算法精确计算TV极小化器作为后处理限制器，经一维、二维测试，可有效抑制振荡且无需调参、保总质量。
AI中文摘要：
守恒律的高阶数值方法在间断附近会产生伪振荡。我们提出通过总变分（TV）去噪步骤对数值解进行后处理以抑制这些振荡，该步骤可降低节点值的总变分。对于一般分析算子，所得离散极小化问题无需满足现有精确最大流算法所需的次模性，因此我们改用微分包含算法精确计算TV极小化器。在一维情形下，该微分包含算法可在有限步内计算出TV极小化器，无需调谐算法参数，且能保持总质量。在二维情形下，我们按维度应用一维算法。我们将该方法作为傅里叶拟谱方法及五阶有限差分格式的后处理限制器，分别对标量守恒律、可压缩欧拉方程和二维不可压缩欧拉方程进行测试。
英文摘要：
High-order numerical methods for conservation laws can generate spurious oscillations near discontinuities. We propose to suppress these oscillations by post-processing the numerical solution with a total variation (TV) denoising step that reduces the total variation of the nodal values. For a general analysis operator, the resulting discrete minimization problem need not satisfy the submodularity property that existing exact max-flow algorithms require, so we instead compute the TV minimizer exactly using a differential inclusion algorithm. In one dimension, the differential inclusion algorithm computes the TV minimizer in finitely many steps, requires no tuning of algorithmic parameters, and preserves the total mass. In two dimensions, we apply the 1D algorithm dimension by dimension. We test the method as a post-processing limiter for Fourier pseudospectral methods and a fifth-order finite difference scheme applied to scalar conservation laws, compressible Euler equations, and the two-dimensional incompressible Euler equations.
