A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter (Communications in Computational Physics, 2020)
This paper presents a new and better suited formulation to implement thelimiting projection to high-order schemes that make use of high-order local reconstructionsfor hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-momentConstrained finite Volume with WENO limiter of 4th order) method, is anextension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative(gradient or slope) at the cell center as an additional constraint for the cell-wise localreconstruction. The gradient is computed from a limiting projection using the WENO(weighted essentially non-oscillatory) reconstruction that is built from the nodal valuesat 5 solution points within 3 neighboring cells. Different from other existing methodswhere only the cell-average value is used in the WENO reconstruction, the presentmethod takes account of the solution structure within each mesh cell, and thus minimizesthe stencil for reconstruction. The resulting scheme has 4th-order accuracy andis of significant advantage in algorithmic simplicity and computational efficiency. Numericalresults of one and two dimensional benchmark tests for scalar and Euler conservationlaws are shown to verify the accuracy and oscillation-less property of thescheme.