A Hermite Method with a Discontinuity Sensor for Hamilton–Jacobi Equations
A Hermite Method with a Discontinuity Sensor for Hamilton–Jacobi Equations
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DOI：
10.1007/s10915-022-01766-2
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发表时间：
2021-05
影响因子：
2.5
通讯作者：
Allen Alvarez Loya;D. Appelö
中科院分区：
文献类型：
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作者：
Allen Alvarez Loya;D. Appelö
提出了一个基于Hermite插值的Hamilton-Jacobi偏微分方程求解器。许多Hamilton-Jacobi方程对梯度有非线性依赖，这导致解的导数不连续，从而导致扭结。我们构建求解器有两个目标：(1)光滑区域的高阶精度和(2)扭结的高分辨率。为了实现这一点，我们使用了带有平滑传感器的赫米特插值。厄米特方法的自由度是张量积泰勒多项式在每个坐标方向上的度数。该方法使用每个节点和维度的自由度，当解是光滑的时候，达到了一个精度的阶。为了获得清晰的扭结分辨率，我们在每个时间步长感知每个单元解的平滑度。如果解是光滑的，我们将不加修改地及时向前推进插值。当我们的方法遇到溶液不光滑的细胞时，它在光滑区域正常进行时，在局部引入人工粘度。我们通过数值实验证明，一旦解在导数中失去连续性，求解器就能很好地捕获扭结，同时在光滑区域实现阶精度。
We present a Hermite interpolation based partial differential equation solver for Hamilton–Jacobi equations. Many Hamilton–Jacobi equations have a nonlinear dependency on the gradient, which gives rise to discontinuities in the derivatives of the solution, resulting in kinks. We built our solver with two goals in mind: (1) high order accuracy in smooth regions and (2) sharp resolution of kinks. To achieve this, we use Hermite interpolation with a smoothness sensor. The degrees-of-freedom of Hermite methods are tensor-product Taylor polynomials of degreemin each coordinate direction. The method usesdegrees of freedom per node ind-dimensions and achieves an order of accuracywhen the solution is smooth. To obtain sharp resolution of kinks, we sense the smoothness of the solution on each cell at each timestep. If the solution is smooth, we march the interpolant forward in time with no modifications. When our method encounters a cell over which the solution is not smooth, it introduces artificial viscosity locally while proceeding normally in smooth regions. We show through numerical experiments that the solver sharply captures kinks once the solution losses continuity in the derivative while achievingorder accuracy in smooth regions.