In this paper, a new fifth-order finite difference mapped unequal-sized weighted essentially non-oscillatory (MUS-WENO) scheme is proposed for solving Hamilton-Jacobi equations in multi-dimensions. This new MUS-WENO scheme uses the same widest spatial stencils as that of the classical same order finite difference WENO scheme [19], and could obtain smaller truncation errors and optimal fifth-order accuracy in smooth regions. The MUS-WENO scheme uses a convex combination of a quartic polynomial with two quadratic polynomials. The linear weights can be artificially set as any positive constants on condition that their summation is one. Together with a new mapping function, associated new mapped nonlinear weights are proposed to reduce the difference with the linear weights. So it is a first time that the new finite difference MUS-WENO scheme with a very small ε could reach optimal accuracy order convergence even near extreme points in smooth regions and avoid spurious oscillations near strong discontinuities when solving Hamilton-Jacobi equations. Generally speaking, the main advantages of such new MUS-WENO scheme comparing with the classical WENO scheme [19] are its good convergence, robustness, efficiency and easy extension to multi-dimensions. Numerical experiments are proposed to show the good performance of this new MUS-WENO scheme.

在本文中，提出了一种新的五阶有限差分映射不等大小加权基本非振荡（MUS-WENO）格式，用于求解多维Hamilton-Jacobi方程。这种新的 MUS-WENO 方案使用与经典的同阶有限差分 WENO 方案 [19] 相同的最宽空间模板，并且可以在平滑区域获得更小的截断误差和最佳的五阶精度。MUS-WENO 方案使用四次多项式与两个二次多项式的凸组合。线性权重可以人为地设置为任何正常数，条件是它们的总和为 1。与新的映射函数一起，提出了相关的新映射非线性权重，以减少与线性权重的差异。即使在平滑区域的极值点附近，ε也可以达到最佳精度阶收敛，并且在求解 Hamilton-Jacobi 方程时避免强不连续性附近的虚假振荡。一般来说，这种新的 MUS-WENO 方案与经典的 WENO 方案 [19] 相比的主要优点是它具有良好的收敛性、鲁棒性、效率和易于扩展到多维。提出了数值实验来证明这种新的 MUS-WENO 方案的良好性能。