A wavelet-Galerkin method for solving population balance equations

From Molecular to Plant‐Scale Modeling of Polymerization Processes: A Digital High‐Pressure Low‐Density Polyethylene Production Paradigm

“…To deal with the above high-dimensionality problem, several numerical methods have been proposed in the literature to reduce the infinite system of differential equations into a loworder system. These can be broadly classified into kinetic lumping methods [19,20], global orthogonal collocation [21,22], method of moments [23][24][25][26][27][28], numerical fractionation methods [29][30][31], discrete weighted Galerkin [32][33][34], orthogonal collocation on finite elements [35,36], and sectional grid methods [15,18,37,38]. The above numerical methods are computationally complex and require special mathematical skills.…”

“…The Galerkin formulations [32,33], although computationally demanding, provide a powerful tool for the prediction of the MWD in a free-radical polymerization system. However, the approximation of the infinite summation terms (e.g., resulting from the termination by combination reaction) requires some special treatment.…”

A 2D wavelet-based spectral finite element method for elastic wave propagation

“…The characteristics of Daubechies wavelets and their usefulness for solving partial differential equations have been discussed in various papers [4][5][6][7][8][9], which has led to their application in the fields of fluid dynamics [5,7], electromagnetics [10,11], dynamics of discrete systems [12][13][14] and damage detection [15,16]. Daubechies wavelets have further been applied for representing the spatial derivative operators in one-dimensional [17] and two-dimensional [18,19] elastic wave propagation formulations, leading to an accurate description of the spatial response.…”

“…The values of the filter coefficients can be computed recursively by combining the two-scale relations, Equations (7) and (8), with the three conditions provided by Equation (9). More details on the algorithm for the computation of the filter coefficients can be found in [7].…”

“…The integration of Equation (15) is carried out using the method proposed in [7,8] for solving Burgers and population balance equations. Incorporating the computed connection coefficients in the present wavelet-Galerkin formulation leads to a coupled system of D Â L (spatial) differential equations (with D representing the spatial dimension of the problem), where the time dependency is accounted for in a discretised sense.…”