Judul: The Application of Method-of-Lines and Runge-Kutta Order 5 on the Problems of Biodegradation of Groundwater Pollutants Using Julia
Abstrak: Masalah biodegradasi pencemar air tanah dapat dimodelkan berupa persamaan diferensial parsial (PDP) adveksi-dispersi-reaksi non-linear. Kompleksnya persamaan ini menuntut digunakannya metode numerik yang efisien untuk memperoleh hampiran solusi. Method-of-Lines yang penulis gunakan untuk memberikan hampiran solusi tersebut, menghasilkan suatu sistem persamaan diferensial biasa (PDB) hasil diskritisasi ruang yang umumnya bersifat kaku dan berukuran sangat besar. Dalam hal ini penulis menggunakan metode Runge-Kutta orde 5 yang dikenal stabil sehingga diharapkan mampu menangani sistem PDB yang kaku. Untuk meningkatkan efisiensi, sistem PDB yang diperoleh penulis formulasikan dalam notasi matriks. Pada contoh kasus pertama, disimulasikan pencemar terkonsentrasi di satu lokasi dengan nutrien dan mikroba tersebar merata di seluruh area. Pada contoh kasus kedua, pencemar dan nutrien tersebar merata di seluruh area, sedangkan mikroba terkonsentrasi di delapan lokasi pada area simulasi tersebut. Hasil simulasi numerik untuk kedua kasus menunjukkan bahwa implementasi notasi matriks memberikan solusi numerik yang sama akurat dan jauh lebih efisien dibanding formulasi non-matriks. Dalam penelitian ini, penulis juga mengimplementasikan sendiri metode Runge-Kutta orde 5 lalu membandingkannya dengan metode Tsit5, implementasi Runge-Kutta orde 5 yang disediakan package DiferensialEquation.jl pada perangkat lunak Julia. Hasil simulasi numerik menunjukkan keduanya memberikan solusi numerik yang sama akurat pada kedua contoh kasus. Namun demikian, metode Tsit5 kurang efisien dibandingkan metode Runge-Kutta orde 5 yang penulis implementasikan., The biodegradation process of groundwater pollutants can be modelled as a non-linear advection-dispersion-reaction partial differential equations (PDE). To solve these equations, it requires the efficient numerical methods to obtain approximate solutions. The method-of-lines which is employed to get the approximate solutions produces a system of ordinary differential equation (ODE) as a result of the space discretization which is generally stiff and very large. In this case, this research uses the fifth-order Runge-Kutta method, which is known as a stable method so that it is expected to handle such a stiff ODE system. To improve its efficiency, the ODE system is converted into the vectorized system. In the first case example, it is simulated that the pollutant is concentrated in one location, while nutrient and microbe are distributed evenly throughout the area. For the second case example, pollutant and nutrient are evenly distributed throughout the area, while microbe is concentrated at eight locations within the simulation area. Numerical results for both cases using non vectorized and vectorized implementation show that their result have similar numerical accuracy outputs, but the vectorized implementation requires much less computing time. This study also compares our implementation of the 5th order Runge-Kutta and the Tsit5 method, the 5th order Runge-Kutta implementation provided by the package of DifferentialEquation.jl in Julia. The numerical simulation result shows that both have similar numerical solution accuracy. However, the Tsit5 method is less efficient than the implemented 5th order Runge-Kutta since it takes far more computing time.
Keyword: Method-of-Lines, Runge-Kutta, biodegradation, advection, dispersion, reaction, Julia