High-order compact difference method for solving the multidimensional population balance equation
Multidimensional population balance equations(MPBE)describe the size distribution of a granular process over two or more intrinsic variables.Since most PBEs lack analytical solutions,computationally expensive high-order or high-resolution(HR)methods are often used to obtain accurate numerical solutions.How to obtain numerical solutions efficiently and accurately is a challenge it faces.To address the above problems,an improved higher-order compact difference(HOCD)method combining dimensional splitting is proposed,which enables the numerical solution with fourth-order accuracy.Dimensional splitting methods are used to split the multidimensional problem into several one-dimensional problems.The split one-dimensional equations are then discretized in space and time to produce the tridiagonal format of the HOCD,which may be solved by using the Tomas algorithm.Variable substitution is also carried out in some cases.Furthermore,the stability was demonstrated by using the von Neumann stability analysis method.Compared with HR methods,the HOCD has higher computational accuracy and computational efficiency without numerical diffusion.The effectiveness of this method is demonstrated by multiple numerical simulations.
population balance equationhigh-order compact differencesize distributiondimensional splittingstability analysisnumerical simulation
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