細胞自動機應用於水體汙染物濃度場模擬之初探

Abstract

為瞭解水體汙染物傳輸的過程，傳統上係以求解平流擴散方程式來模擬水體汙染物傳輸，其中基於Godunov-type的有限體積法(finite volume method, FVM)為相對較廣為使用的數值模式，然其應用於高空間解析度網格時，雖能達到良好的準確度但其計算時間仍然過長。另一方面，應用細胞自動機(Cellular Automata, CA)所發展的模式，以簡單的轉換規則取代過往傳統數值模式中複雜的數值方法，能達到與傳統數值模式相當接近的模擬結果但卻節省大量運算時間的模式，已應用於愈來愈多的領域上。 因此，本研究嘗試以細胞自動機為架構，來發展應用於穩態緩變量流況下之細胞自動機水體汙染物傳輸模式(CA模式)，以簡單之代數方程式來模擬平流與擴散機制，其中平流機制的模擬以各細胞中心點的流速判斷汙染物被動(passive)傳輸的方向，而擴散機制的部分則以濃度為指標來劃定汙染物主動(active)傳輸的方向，並進一步以濃度差為權重值來簡化計算量。為了透過多個平流機制計算方法的優缺點選出一個較適合的方法，此CA模式在平流機制內發展了四個不同的濃度空間規則(upwind、central、QUICK與TVD規則)，再搭配相同之擴散機制模擬方法下，得到CA-upwind、CA-central、CA-QUICK與CA-TVD模式等四個CA模式。 為評估此四個CA模式在準確度與演算效率上的表現，本研究選取四個包含有空間或時間上濃度不連續面之一維或二維穩態緩變量流的水體汙染物傳輸數值案例，將此四個CA模式與Godunov-type FVM模式(FVM模式)進行比較。而在此四個CA模式中，在平流主導下，在具有濃度不連續面的案例下(案例一至三)，CA-upwind模式因有明顯的數值擴散、CA-central與CA-QUICK兩模式因皆有數值震盪等而使此三者準確度不佳，而CA-TVD模式的準確度則幾乎與FVM模式相同；而在案例四中，CA-upwind與CA-central皆有比較明顯的數值擴散，CA-QUICK與CA-TVD則與FVM結果相近。另一方面，在擴散主導下，四個CA模式的準確度則與FVM模式相近或幾乎一樣。至於演算效率部分，不論在平流主導或是擴散主導下，四個CA模式整體而言比FVM模式快約155.0%~246.6 %，且四個CA模式的計算時間差異不大，顯示CA模式均有良好的演算效率。綜合評估下，CA-TVD模式能達到與FVM模式幾乎一樣的準確度但卻能比FVM模式快155.0%~244.1%，可以在穩態緩變量流下水體汙染物傳輸之模擬上取代FVM模式。

Numerical models for predicting solute transport in water bodies are developed by solving the depth-averaged advection-diffusion equations. So far, among various numerical models, the Godunov-type finite volume models are the relatively popular approach. Despite having satisfactory accuracy, Godunov-type finite volume models can introduce high computational demands in high grid resolution. On the other hand, cellular automata (CA) has been applied in more and more fields because it can achieve satisfactory accuracy with significant reduction on computational time by using a set of generic equations instead of tedious numerical procedures of traditional numerical methods. The present study builds a model to simulate solute transport in steady gradually varied flows based on the CA framework (hereinafter in short as the CA model). The CA model simulates the advection and diffusion transport mechanisms with a set of simple algebraic equations. To simulate the advection transport mechanism, the CA model uses the water velocity at the cell center to decide the direction of the transport. For the diffusion transport mechanism, the solute concentration is used as the key to delineate the direction of transport. Moreover, the solute concentration difference is used as the weight to further reduce the calculation involved in simulating the diffusion transport mechanism. To select a more suitable method through the advantages and disadvantages of multiple advection mechanism calculation methods, the CA model adopts four various computation rules (i.e., the upwind, central, QUICK, and TVD rules) to deal with spatial concentration relation. Consequently, there are four CA models, i.e., the CA-upwind, CA-central, CA-QUICK, and CA-TVD models. The accuracy and efficiency of the four CA models are then evaluated through four numerical cases including steady gradually varied flows with spatial/temporal solute concentration discontinuities and compared with a commonly used Godunov-type finite volume model (hereinafter short as the FVM model). From the simulated results, as to the performance of each CA model, concerning the advection-dominated scenarios, in cases with spatial solute concentration discontinuities (Cases 1 to 3), the CA-upwind model has significant numerical diffusions, and both the CA-central and CA-QUICK models have obvious numerical oscillations. Conversely, the CA-TVD model is almost as accurate as the FVM model. In Case 4 with temporal solute concentration discontinuity, both the CA-upwind and CA-central models have relatively obvious numerical diffusions, whereas the simulated results of the CA-QUICK and CA-TVD models are quite similar to the FVM model. In the aspect of the diffusion-dominated scenarios, the accuracy of the four CA modes is similar or almost the same as the FVM model. In terms of efficiency, the four CA models are 155.0%~246.6% faster than the FVM model, which demonstrates the benefit of using the CA framework. Also, it is found that the computational times of the four CA models are close. Comprehensively speaking, the CA-TVD model is found to achieve almost the same accuracy as the FVM model but can be faster than the latter one by 155.0%~244.1%. Thus, the CA-TVD model can replace the FVM model when simulating solute transport in steady gradually varied flows.