Uncertain Grids Method

The most commonly used numerical methods for the solution of PDEs are grid-based methods, notably finite difference (FD) and finite element (FE), and meshless methods, notably particle-in-cell (PIC) and free-Lagrange (FL).  Each of these methods has advantages and disadvantages, but none of them is able to accurately model some of the most difficult problems encountered in advanced scientific research.  The failure of these methods to provide a usable, robust and accurate numerical method for modeling large deformations flows was the catalyst for a new idea – the Uncertain Grids Method.

UGM builds on the advantages of meshless methods, allowing the modeling of large deformations, but corrects the fundamental flaws of these methods, such as the nonmonotonicity and loss of accuracy of derived numerical solutions.

The key to UGM is its approximation of the surface and volume integrals of the unknown functions in a particle as sums of these functions’ values in the points (particles) which are closest to that particle.  To obtain the constitutive equations, one should substitute the Taylor expansion in the corresponding approximation and null the corresponding members.  The obtained relationships should be complemented by the condition of non-negativeness of the coefficients, thereby leading to a linear programming formulation for the coefficients of the gradient approximation.

So each particle has something like its own cloud of particles around it which is used to construct the approximation.  The power of this cloud (equal to the number of particles it contains) can vary without significantly decreasing the quality of the approximation.  The term “particle” is a useful description, but in reality what we have is a volume which is not strictly bounded but about which we know certain physical values (density, energy, velocity, etc.) that are necessary for modeling a specific applied problem.  The unbounded nature of this volume, or cloud of particles, gives “Uncertain Grids Method” its name. 