2007 Fiscal Year Final Research Report Summary
Development of Reliability Evaluation System for Numerical Solutions by Introducing Stochastic Approach and Application to Complicated Fluid Dynamics Simulation
| Project/Area Number |
18540118
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kanazawa University |
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Principal Investigator |
HATAUE Itaru Kanazawa University, Graduate School of Natural Science and Technology, Professor (50218476)
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| Co-Investigator(Kenkyū-buntansha) |
OSHIMA Yoich Kumamoto University, Graduate School of Natural Science and Technology, Professor (20040404)
ITO Shunji Kanazawa University, Graduate School of Natural Science and Technology, Professor (30055321)
OMORI Katsushi Toyama University, Faculty of Human Development, Professor (20110231)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Randomness / Stochastic difference equation / Simulation / Stability / Traffic flow / Macroscopic model / Flux-free |
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| Research Abstract |
For the purpose of development of reliability evaluation system for numerical solutions, we tried to discuss how structure of numerical solutions of a stochastic difference equation changes by the insertion of errors with the random style from the viewpoints of probabilistic approaches. Errors in solving the nonlinear systems are inserted randomly and structure of solutions becomes very complicated We try to investigate the dependence of the structure of numerical solutions on insertion of random errors As a fundamental study, the stochastic differential equation based on the deterministic logistic differential equation and the Lorenz equations are considered. A new approach, sample mean dynamical system (SMDS), is proposed in order to analyze the dependence of the structure of numerical solutions of discretized dynamical system on insertion of random errors and the relation between the size of noise and characteristics of obtained numerical solutions is discussed.
In addition, we appli
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ed them to the issues of real fluid calculation and the problem of traffic jam in order to analyze several factors which govern nonlinear phenomena. First, we tried to discuss the dependence of the structure of numerical solutions of incompressible fluid equations on insertion of random errors in solving simultaneous equations. Dependence of the averaged structure of numerical solutions of fluid simulations on forcibly added random errors are discussed. Next, we give some theoretical considerations on the flux-free finite-element method for the generalized Stokes interface problem arising from the immiscible two-fluid flow problems. In the flux-free finite-element method, the flux constraint is posed as another Lagrange multiplier to keep the zero-flux on the interface. As a result, the mass of each fluid is expected to be preserved at every time step. We fast study the effect of discontinuous coefficients(viscosity and density)on the error of the standard finite element approximations very carefully. Then, the analysis is extended to the flux-free finite element method. As for the problem of traffic jam, the formation of the traffic congestion in two-lane traffic flow is studied. The two-lane macroscopic model using the optimal velocity model which has been introduced in the microscopic model is constructed on the basis of the one-lane model. We adopt different optimal velocity function to each lane and new rules of changing lanes are introduced. Numerical simulations are performed in order to investigate the characteristic phenomena of two-lane traffic flow In particular, we concentrate the discussion about the property of "Synchronized flow', one of the most characteristic phenomena of two-lane traffic flow Furthermore, the fundamental diagrams from the simulations are compared with those observed in a highway. Less
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Research Products
(22 results)
