The Simulation of Nonlinear Transport - Term Paper Example

Name Instructor Subject 21 May 2009 Impact of ‘time stepping schemes’: Incorporation of finite volume method with ‘time stepping schemes’ for nonlinear transport simulation. Abstract This thesis proposal tries to bring out the possibility of the simulation of nonlinear transport by the incorporation of finite volume method with ‘time stepping schemes.’ The favored ‘time stepping scheme’ in this study would be the implicit scheme, though other ‘time stepping schemes’ such as the Crank Nicolson and Explicit schemes will be featured too. The proposals objective range from a matter of aspects, the main ones being the control situation where most time stepping schemes have proved to be a fundamental. The controller is vital where its decision determines the appropriation for a particular time step of either a monolithic scheme or a split scheme, either, known as a fractional step in a high-leveled nonlinear system. This in turn would greatly impact positively in the flow and automation of a nonlinear transport system. Key words: Implicit ‘time stepping scheme’, finite volume method, discretisation, Introduction Nonlinear transport has been in a long time, a standard process. There has been however major revolutions in the current times where innovative scientists have improvised various means in the simulation of nonlinear transport. These have passed previous scientific tests and have been approved. One is the incorporation of ‘finite volume method’ with radial basis methods. Here, a proposal of how the incorporation of ‘finite volume method’ with ‘time stepping schemes’ can simulate nonlinear transport is proposed. Nonlinear transport implementation can be modified in such a manner that time stepping schemes (otherwise referred to as “hybrid”) can be incorporated with ‘finite volume methods’. The incorporation in return solves technicalities in major nonlinear coupled problems. “Finite volume methods” are usually discretization techniques that suit various forms of numerical simulation such as hyperbolic or parabolic scientific laws. It is extensively used but in this case, fluid mechanics is the centre of application (Park and Guvanasen 118). The introduction of these schemes incorporated with finite volumes in nonlinear transport will result in a system entirely based on efficiency and every detailed bit of accuracy. This in turn would result in nonlinear transport having a vast application range. Such would include silylation which is a notably high non-linear system, deeply coupled problematic issues about the diffusion rate of the solvent and the resulting reaction in the deformation of polymers. There are various types of time stepping schemes ranging from Crank-Nicolson time stepping scheme to implicit and explicit ones. This proposal will however center its focus on the implicit time stepping scheme. Implicit time stepping is a ‘high order’ scheme, a compliment that grants it a better grip on matters of stability. It also has a high command on accuracy. This leaves no wonder of its choice as the main time stepping scheme in this proposal (Prashanth and Sanjay 33). Review The critical aspects of research in this thesis proposal will be the time stepping scheme that is to be used. Time stepping schemes are techniques incorporated for the simulation process due to very minimal loss when it comes to accuracy in the simulation of the transport. They also offer improvements that have proved to be significant in the efficiency level of the simulation, a key factor that has led to this proposal. Implicit time stepping in this case has been favored. Implicit time stepping scheme is one of the numerous schemes of time stepping such as Crank Nicolson, Fractional and explicit time stepping schemes. Its superiority over the others in terms of well organized procedural steps was a major approval for it (Luben, Montagnoli and Silva 67). The rest of the proposal ideas are detailed in the rest of the content but having briefed about the key subjects in the content material, I hope that one can have a general idea of this proposal. Research Objectives The objectives of the research in this proposal are multiple. One is to see how time stepping schemes could influence the simulation of nonlinear transport when incorporated with finite volume method. The time stepping scheme in question to be incorporated for the operation is implicit time stepping scheme. For its proven simplified form of operation, we would like to observe the modification that it would result in the simulation of nonlinear transport. Another aim is to scientifically verify if time stepping schemes can really formulate a modus oparendi in which nonlinear transport could gain automation or an improved control system. The mode or methodology of operation in nonlinear transport is also to be observed under the new environment of an incorporated system. The modifications or rather improvements should thus prove beneficial in the simulation of nonlinear transport (Ferziger and Peric 35). All these objectives will clearly be demonstrated in the subtopics of methodology, data interpretation and analysis and finally the outcomes of the application part of the thesis objectives. Research Scenario The scenario in this case is the flow or rather, transport of a liquefied form. The simulation of nonlinear transport is aimed at maintaining a stable and reliable accuracy and also shows improvements. Simulations’ accuracy of such nature, numerical in this case, is highly dependant on space and the temporarity of levels of discretization. The incorporation of the time step scheme, in this case the implicit time stepping scheme, with the finite volume method should thus prove its prowess. This is in order to sustain a satisfactory level of accuracy in the simulation of nonlinear transport. Implicit time stepping scheme is to help in the simulation whereby the medium to be transported is in a situation where forces of capillary and gravity are absent. The time stepping scheme improvises in such a way that it will introduce a system of nonlinear algebra to solve the huge time constants’ differences in the domain (Harvlat 93). Research Methodology The proposal will dwell on the methodology where implicit time stepping scheme will be used. The incorporation of this with finite volume methods should simulate nonlinear transport. Finite volume methods are usually beneficial for fluid flow calculation. This is in the sense that it is good in saving time any available space and number stability. This information is vital in case of a conflict of choice between “finite volume methods” and “finite element method”, whereby finite volume method has proved to be a better choice (Tarck 36). The incorporation of finite volume method with time stepping scheme is thus selected in efforts to simulate nonlinear transport. We will find that these will bring up a workable simulation based from the characters drawn from the two. Implicit is the most considerable scheme that is not temporal thus gives a chance of solving problems with temporal as well as spatial variations. Implicit time stepping scheme is a favored scheme over the explicit time stepping schemes. This is majorly due to the basis of stability where the explicit type contains step sizing restrictions which interfere with stability. It follows a procedural step by step system making it a viable scheme to solve problems with temporal as well as spatial variations. Discretisations of the implicit time stepping scheme able of performing larger time steps are usually preferred in day to day practical calculations. It largely includes a wide range of algebra equations of the particular nonlinear system. These set of algebraic equations are solved at every single time step in order to pacify or rather satisfy the algebra’s particular constraints. This scheme is most favorable in the context of “differential algebraic equations (DAEs)” (Prashanth and Sanjay 820). In its incorporation with finite volume, a sequential fully implicit (SFI) multi-scale finite volume (MSFV) algorithm is the suitable algorithm to be implemented. Previous tests have shown computational and clear efficiency demonstrated by the MSFV algorithm. Implicit scheme efficiency relies heavily on the solvers speed. This is in order to have solutions for the ‘boundary-value’ problems. These problems are solved by temporal discretisation implicitly yielding their corresponding solutions. A combination of Gelerkin scheme in its discontinuous form with implicit time stepping scheme is usually advised for the avoidance of spurious oscillatories. This would create an equation of a temporary semi-discretised form (George 67). All these steps incorporated in this methodology are represented in form of complex mathematical models. These are then solved using the implicit time stepping schemes, incorporated with finite volume method and their outcome models a simulation of nonlinear transport. This should result in accurate simulations that in fact display some improvements such as stability, especially in vicinities where gravity and capillarity are a non-inclusion. Research - Interpretation and Analysis The technique used by implicit time stepping scheme follows a one step at a time technique. This, detailed in the research methodology, ensures stability in the flow that no other time stepping scheme would achieve. This, even though as discovered may entail high cost criterion due to the sophistication involved in the solving of the algebraic equations, a step by step procedure. “Differential algebraic equations (DAEs)” are a preferred selection due to its high nonlinearism combined with sturdy couplings. This is the best situation where implicit time stepping scheme can be applied in the case of nonlinear transport simulation due to its finite steps in the solving of the individual equations. The overall procedure results into a system that simulates nonlinear transport with improvements such stability, accuracy, a sense of automation and a state in which it can function under unfavorable conditions like absence of gravity and low capillarity force. “Implicit finite-volume upwind schemes for solving purely advective transport in the absence of gravity and Capillary forces” (Natvig and Lie 45). Its analysis thus produced the merits and demerits of the thesis proposal. Merits of the ‘implicit time stepping’ scheme 1. The procedure is purely based on accuracy and efficiency due to the step by step procedure that leaves no single problem unsolved. These finite steps are undertaken by well sophisticated solvers –such as the “Newton-Raphson-type nonlinear solvers. ”- that ensure a sufficient level of solution provision. 2. Implicit time stepping scheme, being a ‘high order’ scheme offers stability due to its sequential flow. This beats other schemes in the lower orders and also in its order, constituting it and the explicit time stepping scheme. 3. It follows a procedural criterion of algebraic equations being solved in a step by step format. This ensures fully satisfied constraints in the nonlinear equations. Demerits of implicit time stepping scheme The only certain problem, which is anyway a viable possibility, is the complexity of equation solving at every single step. This procedure eventually makes it a costly venture. Research Outcomes on Proposed Strategies After the strategies and procedures of the detailed above proposal have been made, a new outcome, comprising of an elaborate simulation of nonlinear transport is brought up. This outcome proves its reliability in the cases where there is no mathematical or scientific problem left unturned in the process. This is so because of the detailed and problem-solving specific procedure undertaken by the implicit time stepping scheme (Redelmeier 454). A better working mode where the transport can materialize in situations of nonexistent gravity and capillarity is also achieved. Most nonlinear transport usually wholly depend their movement in conditions of gravity force and where the capillary force applies. The incorporation of time stepping scheme, in this case implicit, thus develop a simulation that enables a wider range of application of nonlinear transport. The availability of problem solvers for the heavily endowed constraints in the algebraic equations however increases the cost of implementation. This is due to the requirement where sophisticated algebra equations have to be solved in a defined procedural order. Conclusion From all the above strategic and procedural steps, we thus create a simulation of nonlinear transport on the basis of incorporating finite volume method with implicit time stepping scheme. This outcome is not a far cry from the nonlinear transport but does a lot in impacting positively in that the improvisations make it a better simulator. This is on the basis of the merit, properly stated on the merits section of the analysis part. Time stepping schemes come in handy in the simulation of nonlinear transport in the basis of control. The controller is vital where its decision determines the appropriation for a particular time step of either a monolithic scheme or a split scheme, either, known as a fractional step in a high-leveled nonlinear system. This in turn would greatly impact positively in the flow and automation of a nonlinear transport system. Implicit time stepping scheme is an invaluable scheme once incorporated with a finite volume method for the simulation of nonlinear transport. This is clearly demonstrated by the brief explanations as stated (Haughney A8). It offers a sequential flow of steps thus making it a more reliable scheme than others like explicit time stepping scheme and the Crank-Nicholson time stepping scheme. The implementation of implicit schemes has thus gained favor among many industries for the major reasons that it is robust: displaying stability has some realistic cost as compared to the efficiency it provides and shows accuracy to levels of being a probable replacant of nonlinear transport. It’s therefore my conclusion that ‘time stepping schemes’ should be incorporated to ‘finite volume methods’ for flow and some kind of automation. This would result in the efficiency and acknowledgeable accuracy of nonlinear transport. Work cited Prashanth K., Vijalapura and Sanjay, Govindjee. An adaptive hybrid time-stepping scheme for highly non-linear strongly coupled problems.  Berkeley: National Science Foundation, 2005. 15 May 2009. Natvig, R. and Lie, K. Fast Computation of Multiphase Flow in Porous Media by Implicit Discontinuous Galerkin Schemes with Optimal Ordering of Elements. San Diego: Apress, 2006. Park, J. and Guvanasen, V. Deportment of Earth and Environmental Sciences. Ontario: Waterloo Inc., 2008. Luben, C., Montagnoli, N. and Silva, R. A generalized alternating-direction implicit scheme for incompressible magnetohydrodynamic viscous flows. California: Rheology Group, 2006. Ferziger, J. H. and Peric, M. Computational Methods for Fluid Dynamics, 2nd ed. Canada: Springer-Verlag , 2001.35 Harvlat, V. Finite Element Methods for Stoles Equations. Berlin-Heidelberg: Springer, 1999. Tarck, G. Numeric and hydrodynamic stability: pressure fluid flow. Garden: Natural History Press. George, T. Application of implicit sub-time stepping to simulate flow and transport in fractured porous media. New York: New York University Society, 2006. Redelmeier, D. “Finite Element Methods.” New England Journal of Medicine 336 (1997): 453- 58 Haughney, Christine. “Time stepping to simulate flow and transport’ Hands.” Wash-ington Post 9 Nov. 2000: A8. References and further reading may be available for this article. To view references and further reading you must purchase this article. Read More