Numerical Solutions of Nonsteady Two-Dimensional Transonic Flows

1979 ◽  
Vol 101 (3) ◽  
pp. 341-347 ◽  
Author(s):  
M. Couston ◽  
J. J. Angelini
Keyword(s):  
Numerical Solutions ◽  
Low Frequency ◽  
Two Dimensional ◽  
Lattice Method ◽  
Alternating Direction Implicit ◽  
Flow Problems ◽  
Present Procedure ◽  
Alternating Direction ◽  
Trailing Edge Flaps ◽  
Doublet Lattice Method

An alternating-direction implicit algorithm is applied to solve an improved formulation of the low-frequency, small-disturbance, two-dimensional potential equation. Linear solutions are presented for oscillating trailing edge flaps, plunging and pitching flat-plate airfoils, and compared with results obtained by a doublet-lattice-method. Nonlinear calculations for both steady and unsteady flow problems are then compared with results obtained by using the complete Euler equations. The present procedure allows one to solve complex aerodynamic problems, including flows with shock waves.

scholarly journals Pedestrian Walking Behavior Revealed through a Random Walk Model

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Hui Xiong ◽  
Liya Yao ◽  
Huachun Tan ◽  
Wuhong Wang
Keyword(s):  
Numerical Experiments ◽  
Flow Simulation ◽  
Two Dimensional ◽  
Pedestrian Flow ◽  
Alternating Direction Implicit Method ◽  
Walking Behavior ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Continuous Time Random Walks ◽  
High Order Compact Scheme

This paper applies method of continuous-time random walks for pedestrian flow simulation. In the model, pedestrians can walk forward or backward and turn left or right if there is no block. Velocities of pedestrian flow moving forward or diffusing are dominated by coefficients. The waiting time preceding each jump is assumed to follow an exponential distribution. To solve the model, a second-order two-dimensional partial differential equation, a high-order compact scheme with the alternating direction implicit method, is employed. In the numerical experiments, the walking domain of the first one is two-dimensional with two entrances and one exit, and that of the second one is two-dimensional with one entrance and one exit. The flows in both scenarios are one way. Numerical results show that the model can be used for pedestrian flow simulation.

scholarly journals Alternating direction implicit type preconditioners for the steady state inhomogeneous Vlasov equation

2017 ◽  
Vol 83 (1) ◽  
Author(s):  
Markus Gasteiger ◽  
Lukas Einkemmer ◽  
Alexander Ostermann ◽  
David Tskhakaya
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Vlasov Equation ◽  
Numerical Solutions ◽  
Splitting Method ◽  
Time Integration ◽  
Computational Effort ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Wide Range

The purpose of the current work is to find numerical solutions of the steady state inhomogeneous Vlasov equation. This problem has a wide range of applications in the kinetic simulation of non-thermal plasmas. However, the direct application of either time stepping schemes or iterative methods (such as Krylov-based methods such as the generalized minimal residual method (GMRES) or relaxation schemes) is computationally expensive. In the former case the slowest time scale in the system forces us to perform a long time integration while in the latter case a large number of iterations is required. In this paper we propose a preconditioner based on an alternating direction implicit type splitting method. This preconditioner is then combined with both GMRES and Richardson iteration. The resulting numerical schemes scale almost ideally (i.e. the computational effort is proportional to the number of grid points). Numerical simulations conducted show that this can result in a speed-up of close to two orders of magnitude (even for intermediate grid sizes) with respect to the unpreconditioned case. In addition, we discuss the characteristics of these numerical methods and show the results for a number of numerical simulations.

scholarly journals Maximum Norm Error Estimates of ADI Methods for a Two-Dimensional Fractional Subdiffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Yuan-Ming Wang
Keyword(s):  
Finite Difference Methods ◽  
Maximum Norm ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Truncation Errors ◽  
Alternating Direction ◽  
Difference Methods ◽  
Subdiffusion Equation ◽  
Adi Methods ◽  
Norm Error

This paper is concerned with two alternating direction implicit (ADI) finite difference methods for solving a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods is provided in the discrete maximum norm. It is shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. Numerical results are given to confirm the theoretical analysis results.

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