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Shallow Liquid Simulation Using Matlab (2001 Neumann) Episode 1

Tham khảo tài liệu 'shallow liquid simulation using matlab (2001 neumann) episode 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | AMATH 581 Homework 2 Shallow Liquid Simulation Erik Neumann 610 N. 65th St. Seattle WA 98103 erikn@MyPhysicsLab.com November 19 2001 Abstract A model of shallow fluid behavior is evaluated using a variety of numerical solving techniques. The model is defined by a pair of partial differential equations which have two dimensions in space and one dimension of time. The equations concern the vorticity w and the stream function b which are related to the velocity field of the fluid. The equations are first discretized in time and space. The time behavior is evaluated using a Runge-Kutta ordinary differential equation solver. The spatial behavior is solved using either Fast Fourier Transform Gaussian Elimination LU Decomposition or iterative solvers. The performance of these techniques is compared in regards to execution time and accuracy. Contents 1 Introduction and Overview 2 2 Theoretical Background 4 2.1 Solving for b - Matrix Method. 5 2.2 Solving for b - FFT Method. 6 2.3 Discretize the Advection-Diffusion Equation. 6 3 Algorithm Implementation and Development 7 3.1 Construction of Matrix A. 8 3.2 Construction of Matrix B. 9 3.3 Pinning the Value of b 1 1 . 10 1 3.4 Comparing Solvers. 11 3.5 An FFT problem. 11 4 Computational Results 11 4.1 Results for various initial conditions . 11 4.2 Running times . 17 4.3 Accuracy of solvers. 17 4.4 Symmetry of Solution. 18 4.5 Time Resolution Needed. 18 4.6 Mesh Drift Instability. 18 5 Summary and Conclusions 21 A MATLAB functions used 22 B MATLAB code 23 B.1 evhump.m . 23 B.2 evrhs.m . 23 B.3 wh.m . 23 B.4 fr.m . 23 B.5 ev2.m. 24 C Calculations 34 1 Introduction and Overview We consider the governing equations associated with shallow fluid modeling. The intended application is the flow of the earth s atmosphere or ocean circulation. The model assumes a 2-dimensional flow with not much movement up or down. Another assumption is that the fluid is shallow ie. that the vertical dimension is much smaller than the horizontal .