1D MLSM and FDM comparison - Revision history

Jureslak: /* Preliminaries */

2017-06-20T15:26:49Z

‎Preliminaries

2017-03-13T11:52:25Z

‎Neumann case

2017-03-13T11:11:55Z

‎Dirichlet case

2017-03-13T10:20:10Z

‎Neumann case

2017-03-13T10:19:56Z

‎Neumann case

2017-03-13T10:09:53Z

2017-03-13T10:09:39Z

2017-03-13T10:07:20Z

2017-03-13T10:05:50Z

2017-03-13T09:59:40Z

‎Neumann case

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 == Preliminaries ==
-When solving such problems one usually makes the disretization of the form <math>f''(x_i) \approx (f_{i-1} -2 f_i + f{i+1}) / h^2</math>,
+When solving such problems one usually makes the disretization of the form <math>f''(x_i) \approx (f_{i-1} -2 f_i + f_{i+1}) / h^2</math>,
-but when writing the system of equations, it is usually written as $f_{i-1} -2 f_i + f{i+1} = h^2 g(x_i)$.
+but when writing the system of equations, it is usually written as $f_{i-1} -2 f_i + f_{i+1} = h^2 g(x_i)$.
 In the general MLSM formulation it is not possible to always do that and therefore we want to know if there are any numerical differences
-between solving $f_{i-1} -2 f_i + f{i+1} = h^2 g(x_i)$ and $(f_{i-1} -2 f_i + f{i+1})_h^2 = g(x_i)$. No significant differences were noted.
+between solving $f_{i-1} -2 f_i + f_{i+1} = h^2 g(x_i)$ and $(f_{i-1} -2 f_i + f_{i+1})_h^2 = g(x_i)$. No significant differences were noted.
 [[File:dir2systems.png|800px]]

← Older revision		Revision as of 11:52, 13 March 2017
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	[[File:neucmp.png\|800px]][[File:neutimecmp.png\|800px]]		[[File:neucmp.png\|800px]][[File:neutimecmp.png\|800px]]
		+
		+	The note on time complexity from the Dirichlet case apllies in this case as well. we can also see that nothing is to be gained from using only single precision.

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 == Dirichlet case ==
 Precision and execution time are summarised in graphs below.
 [[File:dircmp.png|800px]][[File:dirtimecmp.png|800px]]

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 * onesided finite difference <math>f'(0) \approx (f_1 - f_0) / h</math>
-* symmetric finite difference <math>f'(0) \approx (f_1 - f_{-1}) / h</math> and normal discretization in node 0 <math>f''(0)  \approx (f_1 - 2f_0 + f_-1) /h^2</math>
+* symmetric finite difference <math>f'(0) \approx (f_1 - f_{-1}) / h</math> and normal discretization in node 0 <math>f''(0)  \approx (f_1 - 2f_0 + f_{-1}) /h^2</math>
 * onesided double finite difference <math>f'(0) \approx (-3/2 f_0 + 2f_1 -1/2 f_2) / h</math>

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 * onesided finite difference <math>f'(0) \approx (f_1 - f_0) / h</math>
 * symmetric finite difference <math>f'(0) \approx (f_1 - f_{-1}) / h</math> and normal discretization in node 0 <math>f''(0)  \approx (f_1 - 2f_0 + f_-1) /h^2</math>
-* onesided double finite difference <math>f'(0) \approx (-3/2 f_0 + 2f_1 - -1/2 f_2) / h</math>
+* onesided double finite difference <math>f'(0) \approx (-3/2 f_0 + 2f_1 -1/2 f_2) / h</math>
 MLSM with support size $n = 3$ and three polynomial basis functions results in exactly the FDM method with onesided double finite difference approximation.
 [[File:neucmp.png|800px]][[File:neutimecmp.png|800px]]

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 </math>
-were analysed. Theoretically, [https://en.wikipedia.org/wiki/Finite_difference_method FDM] and [[MLSM | Meshless Local Strong Form Method (MLSM)]] should match completely.
+were analysed. Theoretically, [https://en.wikipedia.org/wiki/Finite_difference_method FDM] and [[Meshless Local Strong Form Method (MLSM) | MLSM]] should match completely.
 This is practivaly demonstrated up to certain discretization level.

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 Three explored options are:
-* onsided finite difference <math>f'(0) \approx (f_1 - f_0) / h</math>
+* onesided finite difference <math>f'(0) \approx (f_1 - f_0) / h</math>
 * symmetric finite difference <math>f'(0) \approx (f_1 - f_{-1}) / h</math> and normal discretization in node 0 <math>f'(0)  \approx (f_1 - 2f_0 + f_-1) /h^23</math>
-* onsided double finite difference <math>f'(0) \approx (-3/2 f_0 + 2f_1 - -1/2 f_2) / h</math>
+* onesided double finite difference <math>f'(0) \approx (-3/2 f_0 + 2f_1 - -1/2 f_2) / h</math>
 MLSM with support size $n = 3$ and three polynomial basis functions results in exactly the FDM method with onesided double finite difference approximation.
 [[File:neucmp.png|800px]][[File:neutimecmp.png|800px]]