Fdtxjr

Multi tool use

Why the "ls" command is showing the permissions of files in a FAT32 partition?

How to I force windows to use a specific version of SQLCMD?

What happens if I try to grapple an illusory duplicate from the Mirror Image spell?

Mimic lecturing on blackboard, facing audience

Do you waste sorcery points if you try to apply metamagic to a spell from a scroll but fail to cast it?

How to leave product feedback on macOS?

How to get directions in deep space?

Why do Radio Buttons not fill the entire outer circle?

Overlapping circles covering polygon

Does the Crossbow Expert feat's extra crossbow attack work with the reaction attack from a Hunter ranger's Giant Killer feature?

"Oh no!" in Latin

If Captain Marvel (MCU) were to have a child with a human male, would the child be human or Kree?

Should I assume I have passed probation?

Difference between shutdown options

The Digit Triangles

Deciphering cause of death?

Is there anyway, I can have two passwords for my wi-fi

When and why was runway 07/25 at Kai Tak removed?

Why does a 97 / 92 key piano exist by Bösendorfer?

ContourPlot — How do I color by contour curvature?

Do I have to know the General Relativity theory to understand the concept of inertial frame?

How can I, as DM, avoid the Conga Line of Death occurring when implementing some form of flanking rule?

Can I say "fingers" when referring to toes?

How many people need to be born every 8 years to sustain population?

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