Search

In moving mesh methods, physical PDEs and a mesh equation derived from equidistribution of an error metrics (so-called the monitor function) are simultaneously solved and meshes are dynamically concentrated on steep regions (Lim et al., 2001). However, the simultaneous solution procedure of physical and mesh equations suffers typically from long computation time due to highly nonlinear coupling between the two equations. Moreover, the extended system (physical and mesh equations) may be sensitive to the tuning parameters such as a temporal relaxation factor. It is therefore useful to design a simple and robust moving mesh algorithm in one or multidimension. In this study, we propose a sequential solution procedure including two separate parts: prediction step to obtain an approximate solution to a next time level (integration of physical PDEs) and regriding step at the next time level (mesh generation and solution interpolation). Convection terms, which appear in physical PDEs and a mesh equation, are discretized by a WENO (Weighted Essentially Non-Oscillatory) scheme under the consrvative form. This sequential approach is to keep the advantages of robustness and simplicity for the static adaptive grid method (local refinement by adding/deleting the meshes at a discrete time level) as well as of efficiency for the dynamic adaptive grid method (or moving mesh method) where the number of meshes is not changed. For illustration, a phase change problem is solved with the decomposition algorithm.

Publisher Summary This chapter discusses moving finite difference method combined with Weighted Essentially Non-Oscillatory (WENO) scheme for the dynamic simulation of batch crystallization processes described by hyperbolic-like Population Balance Equation (PBE) with a discontinuous initial condition. The accurate and stable WENO scheme shows an improvement of numerical results over conventional discretization methods (backward or central) on fixed grids as well as on moving grids. Owing to the moving grid method with the WENO scheme, which tracks well a steep front or shock, the physical model of PBEs could be numerically represented more exactly. Numerical results are compared with experimental results for the crystallization process of the potassium sulfate (K 2 SO 4 /H 2 O). The new approach is considered as an efficient numerical solution procedure for the verification of models described by hyperbolic-like Partial Differential Equations (PDEs).

For tracking a shock or steep moving front in the convective Partial Differential Algebraic Equations (PDAEs), some spatial discretization methods such as central, upwinding, Essentially Non-Oscillatory (ENO) and Weighted Essentially Non-Oscillatory (WENO) schemes, and the moving grid techniques are examined respectively and simultaneously with regards to accuracy, temporal performance and stability. The uniform fixed-grid method is efficient owing to relatively short CPU time. However, the solution of the Fixed Stencil (FS) approach such as the central and upwinding spatial discretization methods is unstable. Many mesh points are needed for all of the fixed grid methods to obtain enough solution accuracy. In the moving grid techniques, stability of the solution is enhanced even at small mesh numbers but it is prohibitive because of the addition of a complex and nonlinear mesh equation to solved PDAEs. The combination of the ENO/WENO schemes (based on an adaptive stencil idea) with moving grid techniques gives a stable and accurate solution event at small mesh numbers and is efficient to track a moving shock front.

This study attempts to address efficient implementation of the normal boundary intersection (NBI) method for the tradeoff between economics and pollution prevention with appropriate analysis methods. For the multiobjective optimization problem (MOOP), a noninferior solution curve (or Pareto curve) is efficiently obtained by using an enhanced NBI method combined with the SWOF (summation of weighted objective functions) method within a chemical process simulator. Then, the Pareto curve is analyzed by the ideal compromise solution set, sensitivity and elasticity analyses that give the decision basis between the conflicting objectives to the decision maker. Through the combined use of the SWOF and NBI methods, the upper and lower values of the ideal compromise set are exactly located. The total calculation time for the entire Pareto curve can be reduced owing to evenly distributed Pareto points.