**Author details**

*<sup>μ</sup>*½ � *<sup>k</sup>* <sup>¼</sup> <sup>1</sup> Δ*t dρ du u*½ � *<sup>k</sup>*

*Recent Advances in Numerical Simulations*

**7. Conclusion and future works**

have been implemented very efficiently.

steps of the simulation again.

**Acknowledgements**

DTU-Biosustain.

**110**

� *df du u*½ � *<sup>k</sup>*

reaction equation where the latter was explained in this chapter.

, *<sup>q</sup>*½ � *<sup>k</sup>* <sup>¼</sup> *<sup>μ</sup>*½ � *<sup>k</sup> <sup>u</sup>*½ � *<sup>k</sup>* <sup>þ</sup> *<sup>f</sup>*

Having bounded derivatives and a good starting point *u*½ � <sup>0</sup> (which is normally chosen *un*), the resulted sequence *u*½ � *<sup>k</sup>* in Eq. (27) converges quadratically to *un*þ1, the solution of Eq. (26). Note that (26) is a diffusion–reaction equation which is completely explained how to solve and (core) codes were provided, as well.

In this introductory chapter we presented the framework of FEM in brief (but effective) and implemented a numerical solver for diffusion–advection-reaction equation. Accumulation of different terms and setting boundary conditions correctly as well as evaluating of definite integrals without numerical integration were explained and their codes were also given. Finally we showed that nonlinear parabolic equations can be solved by combining of Newton method and diffusion–

However, to have a reliable and efficient simulator several topics and challenges should be addressed. Assuming correct mathematical model, mesh generation and assigning (measured or suggested) values to the coefficients of the problem are very important to make close the numerical model and its solution to the real problem [4, 5]. In a real problem, 2D and 3D elements appear together. For example 2D elements are employed to model faults in a 3D reservoir. Moreover, the generated mesh is normally unstructured and different types of elements are included in the mesh. Hence to assembly the linear system, traversing elements and nodes should

Solving linear systems is normally the most time consuming part of the simulation and efficient implementation of linear solvers particularly in parallel machines, are of great interest [6]. Multigrid and multiscale methods, particularly their algebraic form, have attracted interest to solve large scale linear systems since they have shown good scalability in addition to resolving low frequency parts of the error in solving linear systems [7, 8]. Standard Galerkin method that we used in this chapter

is not a conservative scheme hence modifications or other methods such as discontinues Galerkin method would be necessary to solve problems in computational fluid dynamics [9, 10]. Proposing and implementing of advanced numerical algorithms to linearize and solve a system of nonlinear coupled initial-boundary value problem in a large scale domain become necessary. At last comparison with real data (if available) and quantifying uncertainties might force us to revisit all

Hani Akbari would like to thank Prof. Lars K. Nielsen, scientific director at Novo Nordisk Foundation Center for Bio-Sustainability. Hani Akbari is grateful to

DTU-Biosustain since this work was completed when he was a postdoc at

½ � *<sup>k</sup>* � <sup>1</sup> Δ*t*

*ρn:* (28)

Hani Akbari Shamim Institute of HPC, Scientific Visualization and Machine Learning, Mashhad, Iran

\*Address all correspondence to: hani.akbari@shamimhpc.ir

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
