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stability for 2D crank-nicolson scheme for heat equation

Weak solution for Burgers' equationParticular answer to a differential Equation Involving Delta function and Heaviside without LaplaceStability of advection equationLax-Wendroff method for linear advection - Stability analysisCrank-Nicolson for coupled PDE'sUniqueness/stability for heat equations with complicated initial-boundary conditionsCrank Nicolson with variable diffusion coefficient in space and timeIteration step of the Crank–Nicolson schemeUnderstanding a basic scheme for the heat equationCalculating convergence order of numerical scheme for PDEChecking the stability of finite difference schemesStability of numerical scheme for heat equation

0

$begingroup$

We have parabolic 2D pde

beginalign*
v_t &= nu (v_xx + v_yy) + F(x,y,t), ; ; ; (x,y) in R, ; t >0 \
v(x,y,t) &= g(x,y,t), ; ; texton ; partial R, ; t>0 \
v(x,y,0) &= f(x,y), ; ; (x,y) in overlineR \
endalign*

We want to study the stability of scheme

beginalign*
left(1 - frac r_x2 delta_x^2 - frac r_y2 delta_y^2 right) u_jk^n+1 &= u_jk^n + frac Delta t2 (F_jk^n + F_jk^n+1 ) + frac r_x 2 delta_x^2 u_jk^n + frac r_y 2 delta_y^2 u_jk^n\
u_0k^n &= g(0,k Delta y, n Delta t ) \
u_M_x k^n &= g(1, k Delta y, n Delta t) \
u_j 0 ^n &= g(j Delta x, 0, n Delta t) \
u_j M_y ^n &= g(j Delta x, 1 , n Delta t) \
endalign*

the Crank-Nicolson scheme. It is supposed to be uncondionally stable. MY question is,

Do we just need to apply discrete von neumann criteria
$$ u_jk^n = xi^n e^ijp pi Delta x + i k q pi Delta y $$

with exclusion of $F$ source term then get and equation for $xi$

so all we need to find is conditions on $r_x, r_y$ that makes $|xi | leq 1$ which is the necessary condition for stability.

IS this correct?

pde numerical-methods

share|cite|improve this question

asked Mar 14 at 6:18

Mikey SpivakMikey Spivak

382215

$endgroup$

add a comment | 

0

$begingroup$

We have parabolic 2D pde

beginalign*
v_t &= nu (v_xx + v_yy) + F(x,y,t), ; ; ; (x,y) in R, ; t >0 \
v(x,y,t) &= g(x,y,t), ; ; texton ; partial R, ; t>0 \
v(x,y,0) &= f(x,y), ; ; (x,y) in overlineR \
endalign*

We want to study the stability of scheme

beginalign*
left(1 - frac r_x2 delta_x^2 - frac r_y2 delta_y^2 right) u_jk^n+1 &= u_jk^n + frac Delta t2 (F_jk^n + F_jk^n+1 ) + frac r_x 2 delta_x^2 u_jk^n + frac r_y 2 delta_y^2 u_jk^n\
u_0k^n &= g(0,k Delta y, n Delta t ) \
u_M_x k^n &= g(1, k Delta y, n Delta t) \
u_j 0 ^n &= g(j Delta x, 0, n Delta t) \
u_j M_y ^n &= g(j Delta x, 1 , n Delta t) \
endalign*

the Crank-Nicolson scheme. It is supposed to be uncondionally stable. MY question is,

Do we just need to apply discrete von neumann criteria
$$ u_jk^n = xi^n e^ijp pi Delta x + i k q pi Delta y $$

with exclusion of $F$ source term then get and equation for $xi$

so all we need to find is conditions on $r_x, r_y$ that makes $|xi | leq 1$ which is the necessary condition for stability.

IS this correct?

pde numerical-methods

share|cite|improve this question

asked Mar 14 at 6:18

Mikey SpivakMikey Spivak

382215

$endgroup$

add a comment | 

$begingroup$

We have parabolic 2D pde

beginalign*
v_t &= nu (v_xx + v_yy) + F(x,y,t), ; ; ; (x,y) in R, ; t >0 \
v(x,y,t) &= g(x,y,t), ; ; texton ; partial R, ; t>0 \
v(x,y,0) &= f(x,y), ; ; (x,y) in overlineR \
endalign*

We want to study the stability of scheme

beginalign*
left(1 - frac r_x2 delta_x^2 - frac r_y2 delta_y^2 right) u_jk^n+1 &= u_jk^n + frac Delta t2 (F_jk^n + F_jk^n+1 ) + frac r_x 2 delta_x^2 u_jk^n + frac r_y 2 delta_y^2 u_jk^n\
u_0k^n &= g(0,k Delta y, n Delta t ) \
u_M_x k^n &= g(1, k Delta y, n Delta t) \
u_j 0 ^n &= g(j Delta x, 0, n Delta t) \
u_j M_y ^n &= g(j Delta x, 1 , n Delta t) \
endalign*

the Crank-Nicolson scheme. It is supposed to be uncondionally stable. MY question is,

Do we just need to apply discrete von neumann criteria
$$ u_jk^n = xi^n e^ijp pi Delta x + i k q pi Delta y $$

with exclusion of $F$ source term then get and equation for $xi$

so all we need to find is conditions on $r_x, r_y$ that makes $|xi | leq 1$ which is the necessary condition for stability.

IS this correct?

pde numerical-methods

share|cite|improve this question

asked Mar 14 at 6:18

Mikey SpivakMikey Spivak

382215

$endgroup$

share|cite|improve this question

asked Mar 14 at 6:18

Mikey SpivakMikey Spivak

382215

share|cite|improve this question

asked Mar 14 at 6:18

Mikey SpivakMikey Spivak

382215

asked Mar 14 at 6:18

Mikey SpivakMikey Spivak

382215

asked Mar 14 at 6:18

Mikey SpivakMikey Spivak

382215