The Structural Method for Partial Differential Equations
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The order of convergence of Finite Difference methods is limited by the stencil size. For exemple, a 3-point scheme cannot be more than 2nd order accurate. Taking a larger stencil allows for higher order schemes, at the cost of lower spectral resolution and some special care for boundary points. The Structural Method overcomes those limitations by introducing state variables for the differential operators involved in the PDE. Thanks to the local information of the state variables, such a scheme can be of order 6th while keeping a compact 3-point stencil, thus having nice spectral properties. In the presentation, we will first introduce the structural method, comparing it to classical finite differences. We will then explain how to automatically compute the structural equations given a stencil and state variables. The properties of the structural method will finally be shown on simulations of hyperbolic PDEs (e.g. Advection, Burger’s, Shallow water, ...).
