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1- Discretize the problem using finite difference of second order pf PDE which can be obtained by Taylor series expansion and 2nd order 3 points PDE at time p+1

2- The stable and highly accurate BRIAN algorithm is well suited in parallel implementation in this study.

3- Speedup is the ratio of time taken for a program to execute in one processor to time taken to execute in many processors. Efficiency is a measure of processor utilization. High value of temporal performance shows better performance of parallel algorithm. Granularity is the ratio of the amount of computation to the ratio of the amount of communication within a parallel algorithm implementation

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This is  a  discretization process of  PDE using FDM . Solving FDM using parallel  BRIAN algorithm. Some indicators are used to investigate the performance of the parallel BRIAN algorithm 

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This is  a  discretization process of  PDE using FDM . Solving FDM using parallel  BRIAN algorithm. Some indicators are used to investigate the performance of the parallel BRIAN algorithm 

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Step-by-Step Discretization:

1. Discretize the Spatial Domain:

Discretize r θ, and z into grid points.

2. Finite difference approximation

Use central differences for spatial derivatives and forward differences for the time derivative.

3. Substitute Finite Differences into the PDE

4. Solve for U^(n+1)

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Parallel AGE solver of multidimensional PDE modeling for Thermal Control of Laser Beam on Cylindrical Glass

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 Do you think this problem is considered as complex problem 

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 Do you think this problem is considered as complex problem 

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1. This is a 3D problem, a typical high-dimensional real-world case.

2. PDE is more complex than ODE

3. A lot of parameters are included in the problem(based on the article) - increased complexity

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 (4). Execution time, speed up, effectiveness and temporal performance increase as the number of processor grow up. BRIAN methods is different results from the other methods using (70X70X70) sizes of matrices. DOUGLAS has different results for 1 dimensional problem. 

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 (4). Execution time, speed up, effectiveness and temporal performance increase as the number of processor grow up. BRIAN methods is different results from the other methods using (70X70X70) sizes of matrices. DOUGLAS has different results for 1 dimensional problem. 

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Describe step by step  on how you discretise this PDE with cylindrical coordinate. List the domain and range for this 3D problem. 

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Describe step by step  on how you discretise this PDE with cylindrical coordinate. List the domain and range for this 3D problem. 

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The performance of Parallel computing for solving PDE above can be investigated by some indicators such as execution time, speedup, efficiency , effectiveness and temporal performance. Explain your understanding 

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The performance of Parallel computing for solving PDE above can be investigated by some indicators such as execution time, speedup, efficiency , effectiveness and temporal performance. Explain your understanding 

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Can your describe the differences all elements of performance indicators between figure (4-9) and figure ( 10-13)