Generalized Minmod Piecewise Linear Reconstruction in both 1-D and 2-D Cases

In this article, we present a new method for reconstructing functions based on piecewise linear approximations. Our approach is inspired by the minmod function, which is widely used in numerical analysis but has limitations when dealing with complex or irregularly-spaced data. By introducing a generalized version of the minmod function, we can handle non-uniform grids and account for second-order accuracy in our reconstructions.
The key idea behind our method is to use slopes computed at cell interfaces to construct a piecewise linear approximation of the original function. These slopes are adjusted using a modified minmod function that takes into account the non-uniform grid and provides improved accuracy. The resulting reconstruction is smooth and continuous, with no oscillations or discontinuities at the grid points.
Our approach is based on the concept of minimizing the maximum deviation of the reconstructed function from its original values over a given interval. This is achieved by adjusting the slope of the piecewise linear approximation in such a way that the maximum deviation is minimized. The resulting reconstruction provides an accurate representation of the original function, with improved accuracy compared to traditional piecewise linear methods.
The main advantage of our method is its ability to handle complex or irregularly-spaced data without sacrificing accuracy. Unlike traditional piecewise linear methods, which rely on uniform grids and can be prone to oscillations or discontinuities, our approach provides a smooth and continuous reconstruction that accurately represents the original function.
In summary, we have introduced a generalized minmod piecewise linear reconstruction method that provides an accurate and efficient way of reconstructing functions based on non-uniform grids. Our approach leverages the concept of minimizing maximum deviation to adjust the slope of the piecewise linear approximation, resulting in a smooth and continuous reconstruction that captures the essence of the original function.