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A simple implicit scheme for the heat equation was proposed by Laasonen (1949). The algorithm for this scheme is u+¹-u At α u-2u+¹+u (Ax)² If we make use of the central-difference operator 8uu1-2u+u₁-1 we can rewrite Equation 4.92 in the simpler form: u-u At - ) = [/2a²4² + ut – Qlixx ² At+ α(4x)² 12 8u+1 (Ax)² This scheme has first-order accuracy with a T.E. of O[(At)², (Ax)² ] and is unconditionally stable. Upon examining Equation 4.93, it is apparent that a tridiagonal system of linear algebraic equations must be solved at each time level n +1. The modified equation for this scheme is + [10²³(At)² + 1 20²A² (Ax)² + 360° 300 (4x)* Ju .... (4.92) *** (4.93) (4.94)

A simple implicit scheme for the heat equation was proposed by Laasonen (1949). The algorithm for this scheme is u+¹-u At α u-2u+¹+u (Ax)² If we make use of the central-difference operator 8uu1-2u+u₁-1 we can rewrite Equation 4.92 in the simpler form: u-u At - ) = [/2a²4² + ut – Qlixx ² At+ α(4x)² 12 8u+1 (Ax)² This scheme has first-order accuracy with a T.E. of O[(At)², (Ax)² ] and is unconditionally stable. Upon examining Equation 4.93, it is apparent that a tridiagonal system of linear algebraic equations must be solved at each time level n +1. The modified equation for this scheme is + [10²³(At)² + 1 20²A² (Ax)² + 360° 300 (4x)* Ju .... (4.92) *** (4.93) (4.94)

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Derive from the 4.93 equation to the 4.94 equation

A simple implicit scheme for the heat equation was proposed by Laasonen (1949). The algorithm for this scheme is u+¹-u At Ut αuxx = u+1-2u+¹+₁+1 = α- If we make use of the central-difference operator - we can rewrite Equation 4.92 in the simpler form: u₁¹uj At 8u =u+1-2u+u₁_₁ (Ax)² a(Ax)² 12 = α 8u+1 (4x)² This scheme has first-order accuracy with a T.E. of O[(At)², (Ax)² ] and is unconditionally stable. Upon examining Equation 4.93, it is apparent that a tridiagonal system of linear algebraic equations must be solved at each time level n +1. The modified equation for this scheme is Uxxxx + [ { α² (At)² + 1 2 α²³At(^x)² - + 12 ___a (4x) *] ¹.* 360 (4.92) +.. (4.93) (4.94)
Transcribed Image Text:A simple implicit scheme for the heat equation was proposed by Laasonen (1949). The algorithm for this scheme is u+¹-u At Ut αuxx = u+1-2u+¹+₁+1 = α- If we make use of the central-difference operator - we can rewrite Equation 4.92 in the simpler form: u₁¹uj At 8u =u+1-2u+u₁_₁ (Ax)² a(Ax)² 12 = α 8u+1 (4x)² This scheme has first-order accuracy with a T.E. of O[(At)², (Ax)² ] and is unconditionally stable. Upon examining Equation 4.93, it is apparent that a tridiagonal system of linear algebraic equations must be solved at each time level n +1. The modified equation for this scheme is Uxxxx + [ { α² (At)² + 1 2 α²³At(^x)² - + 12 ___a (4x) *] ¹.* 360 (4.92) +.. (4.93) (4.94)
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