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Proving consistency of 1D heat equation

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$begingroup$

I have a text that defines consistency as : $lim_k to 0 frac1k ||R_ku(t)-u(t+Delta t)||$

Where $k = Delta t$, and $R_k$ is a finite difference operator.

The book then says it verifies consistency of the $1$D heat equation like this :

$R_ku-u^j+1_l=-lambda u_l+1^j-(1-2lambda)u_l^j-lambda u_l-1^j$

The author expands all of these terms with Taylor expansion and shows that as $K rightarrow 0$ the expression goes to zero.

How can this prove consistency when the definition of consistency has a $frac1k$ term that the above expression does not have?

Incase it is important, the heat equation used is $fracpartial upartial t=alpha fracpartial upartial x$, and $lambda = alpha fracDelta tDelta x^2$. The book says they are representing the truncation error, "the error between the exact process and the discrete processes, in terms of the Taylor expansion and then reduce the error to one involving only the remainder terms."

finite-differences

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asked Mar 15 at 4:51

FrankFrank

17610

$endgroup$

add a comment | 

0

$begingroup$

I have a text that defines consistency as : $lim_k to 0 frac1k ||R_ku(t)-u(t+Delta t)||$

Where $k = Delta t$, and $R_k$ is a finite difference operator.

The book then says it verifies consistency of the $1$D heat equation like this :

$R_ku-u^j+1_l=-lambda u_l+1^j-(1-2lambda)u_l^j-lambda u_l-1^j$

The author expands all of these terms with Taylor expansion and shows that as $K rightarrow 0$ the expression goes to zero.

How can this prove consistency when the definition of consistency has a $frac1k$ term that the above expression does not have?

Incase it is important, the heat equation used is $fracpartial upartial t=alpha fracpartial upartial x$, and $lambda = alpha fracDelta tDelta x^2$. The book says they are representing the truncation error, "the error between the exact process and the discrete processes, in terms of the Taylor expansion and then reduce the error to one involving only the remainder terms."

finite-differences

share|cite|improve this question

asked Mar 15 at 4:51

FrankFrank

17610

$endgroup$

add a comment | 

$begingroup$

I have a text that defines consistency as : $lim_k to 0 frac1k ||R_ku(t)-u(t+Delta t)||$

Where $k = Delta t$, and $R_k$ is a finite difference operator.

The book then says it verifies consistency of the $1$D heat equation like this :

$R_ku-u^j+1_l=-lambda u_l+1^j-(1-2lambda)u_l^j-lambda u_l-1^j$

The author expands all of these terms with Taylor expansion and shows that as $K rightarrow 0$ the expression goes to zero.

How can this prove consistency when the definition of consistency has a $frac1k$ term that the above expression does not have?

Incase it is important, the heat equation used is $fracpartial upartial t=alpha fracpartial upartial x$, and $lambda = alpha fracDelta tDelta x^2$. The book says they are representing the truncation error, "the error between the exact process and the discrete processes, in terms of the Taylor expansion and then reduce the error to one involving only the remainder terms."

finite-differences

share|cite|improve this question

asked Mar 15 at 4:51

FrankFrank

17610

$endgroup$

share|cite|improve this question

asked Mar 15 at 4:51

FrankFrank

17610

share|cite|improve this question

asked Mar 15 at 4:51

FrankFrank

17610

asked Mar 15 at 4:51

FrankFrank

17610

asked Mar 15 at 4:51

FrankFrank

17610