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70 3. INVESTIGATING THE PRECISION OF PDE SOLVER ARCHITECTURES ON FPGAS determine the optimal computational precision of the advection equation solver architecture. 3.1 The Advection Equation For the test the simple advection equation (3.1) is selected, where the analytical solution is known and can be easily generated. ∂u ∂t +c∂u =0 (3.1) ∂x where t denotes time, u is conserved property, c is the advection speed. The initial condition is a proper periodical function with the property of: u(x,t 0 )=u 0 (x) (3.2) the range of the function is from 0 to 1. Periodic boundary condition are used on the boundaries. With this initial conditions the architecture can easily be tested, because of the periodicity, the result should be in the same range as the initial condition. The analytical solution is ũ(x,t)=u 0 (x−c(t−t 0 )) (3.3) The numerical approximation of the advection equation is not easy, especially if the initial condition u 0 is discontinuous function. Its structure and solution method is similar to those equations which used in Fluid Flow simulation [55]. This is a good starting equation which helps us investigating the precision of the arithmetic unit on FPGA to reach the predefined accuracy of the solution. However this chapter deals only with the one-dimensional case, and this 1D can not be used in real life applications, it gives us experience for the two- and three dimensional cases. 3.2 Numerical Solutions of the PDEs In order to solve the continuous space-time PDE the equations has to be discretized. Since logically structured arrangement of data is fundamental for the
3.2 Numerical Solutions of the PDEs 71 efficient operation of the FPGA based implementations, we consider explicit finite volume discretizations of the governing equations over structured grids employing a simple numerical flux function. In the following we recall the details of the considered first- and second-order schemes. 3.2.1 The First-order Discretization The simplest algorithm we consider is first-order both in space and time. Application of the finite volume discretization method leads to the following semi-discrete form of governing equations dU i dt = − 1 ∑ F f , (3.4) V i,j where the summation is meant for the two faces of cell i, F f is the flux tensor evaluated at face f. Face f separates cell L (left) and cell R (right). In order to stabilize the solution procedure, artificial dissipation has to be introduced into the scheme. Following the standard procedure, this is achieved by replacing the physical flux tensor by the numerical flux function F N containing a dissipative stabilization term. The finite volume scheme is characterized by the evaluation of F N , which is the function of both U L and U R . In the paper we apply the simple and robust Lax-Friedrichs numerical flux function defined as f F N = F L + F R 2 − ¯c·UR − U L 2 and bar labels speeds computed at the following averaged state (3.5) Ū = U L + U R . (3.6) 2 In equation (3.5) c is the velocity component, F L =F x (U L ), F R =F x (U R ). A vast amount of experience has shown that these equations provide a stable discretization of the governing equations if the time step satisfies the Courant– Friedrichs–Lewy condition (CFL condition): u·∆t ≤c (3.7) ∆x where u is the velocity, ∆t is the time-step, ∆x is the length interval.
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An Optimal Mapping of Numerical Sim
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Acknowledgements It is not so hard
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Contents 1 Introduction 1 1.1 Cellu
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CONTENTS iii 4.3.3 Operating Steps
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vi LIST OF FIGURES 2.4 Performance
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List of Tables 2.1 Comparison of di
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xii LIST OF ABBREVIATIONS FPE Falco
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Chapter 1 Introduction Due to the r
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3 can be represented in arbitrary t
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1.1 Cellular Neural/Nonlinear Netwo
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1.2 Cellular Neural/Nonlinear Netwo
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1.3 CNN-UM Implementations 9 the Lo
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1.3 CNN-UM Implementations 11 modif
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1.4 Field Programmable Gate Arrays
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Page 39 and 40: 1.4 Field Programmable Gate Arrays
Page 47 and 48: 1.5 IBM Cell Broadband Engine Archi
Page 55 and 56: Chapter 2 Mapping the Numerical Sim
Page 57 and 58: 2.1 Introduction 37 18 Load instruc
Page 59 and 60: 2.1 Introduction 39 Instruction his
Page 61 and 62: 2.1 Introduction 41 2E+07 1E+07 9E+
Page 63 and 64: 2.1 Introduction 43 Main memory 8 t
Page 65 and 66: 2.1 Introduction 45 6 4.5 3 Speedup
Page 67 and 68: 2.1 Introduction 47 2.50E+07 1.88E+
Page 69 and 70: 2.2 Ocean model and its implementat
Page 71 and 72: 2.2 Ocean model and its implementat
Page 73 and 74: 2.3 Computational Fluid Flow Simula
Page 89: Chapter 3 Investigating the Precisi
Page 93 and 94: 3.3 Testing Methodology 73 In fixed
Page 95 and 96: 3.4 Properties of the Arithmetic Un
Page 97 and 98: 3.4 Properties of the Arithmetic Un
Page 99 and 100: 3.5 Results 79 in the L2 cache of t
Page 101 and 102: 3.5 Results 81 1E-02 1E-03 Error 1E
Page 103 and 104: 3.5 Results 83 arithmetic unit requ
Page 105: 3.6 Conclusion 85 point and the 40
Page 108 and 109: 4. IMPLEMENTING A GLOBAL ANALOGIC P
Page 130 and 131: 110 5. SUMMARY OF NEW SCIENTIFIC RE
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References The author’s journal p
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5.2 Application of the results 123
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