d*********g
发帖数: 49 | 1 做随即控制问题做到一个高维的两次parabolic type PDE,想要用有限差分做一个
数值解试试。现在两次导数前面的系数有state variable, 符号不定，导致
一般做法下得到的finite difference scheme没有单调性。不知道这种
情况是不是还有希望，请教熟悉的同学推荐一些书或者论文。 | d*********g
发帖数: 49 | 2 Thank you, I will take a look at that.
My question is how to properly discretize this equation, so that
the scheme I get has certain monotonicity. | m****n
发帖数: 45 | 3
What is "monotonicity" for a finite difference scheme?
【在 d*********g 的大作中提到】
: 做随即控制问题做到一个高维的两次parabolic type PDE,想要用有限差分做一个
: 数值解试试。现在两次导数前面的系数有state variable, 符号不定，导致
: 一般做法下得到的finite difference scheme没有单调性。不知道这种
: 情况是不是还有希望，请教熟悉的同学推荐一些书或者论文。
| d*********g
发帖数: 49 | 4 因为我是做viscosity solution, 要求这个scheme, 比如u(t,x)=(Tu)(t-delta,dot)
这个operator T是单调的。好像有限元里面有什么proper还是什么性质的讲法，
有点类似，不好意思，我一知半解地说。
【在 m****n 的大作中提到】
:
: What is "monotonicity" for a finite difference scheme?
| c*******e
发帖数: 8624 | 5 你把问题贴出来吧,包括bc,
finite difference和finite element两回事
fe里面的数学背景要高一个数量级
【在 d*********g 的大作中提到】
: 因为我是做viscosity solution, 要求这个scheme, 比如u(t,x)=(Tu)(t-delta,dot)
: 这个operator T是单调的。好像有限元里面有什么proper还是什么性质的讲法，
: 有点类似，不好意思，我一知半解地说。
| d*********g
发帖数: 49 | 6 Thank you, I will take a look at that.
My question is how to properly discretize this equation, so that
the scheme I get has certain monotonicity. | w**a
发帖数: 1024 | 7 i didn't fully understand your problem,
is your problem same to the 'upwind f.d. scheme' in viscosity solution
or 'entropy 'condition?
【在 d*********g 的大作中提到】
: 因为我是做viscosity solution, 要求这个scheme, 比如u(t,x)=(Tu)(t-delta,dot)
: 这个operator T是单调的。好像有限元里面有什么proper还是什么性质的讲法，
: 有点类似，不好意思，我一知半解地说。
| w**a
发帖数: 1024 | 8 i didn't fully understand your problem,
is your problem same to the 'upwind f.d. scheme' in viscosity solution
or 'entropy 'condition?
【在 d*********g 的大作中提到】
: 因为我是做viscosity solution, 要求这个scheme, 比如u(t,x)=(Tu)(t-delta,dot)
: 这个operator T是单调的。好像有限元里面有什么proper还是什么性质的讲法，
: 有点类似，不好意思，我一知半解地说。
| w**a
发帖数: 1024 | 9 i didn't fully understand your problem,
is your problem same to the 'upwind f.d. scheme' in viscosity solution
or 'entropy 'condition?
【在 d*********g 的大作中提到】
: 因为我是做viscosity solution, 要求这个scheme, 比如u(t,x)=(Tu)(t-delta,dot)
: 这个operator T是单调的。好像有限元里面有什么proper还是什么性质的讲法，
: 有点类似，不好意思，我一知半解地说。
| t**********r
发帖数: 256 | 10 谱方法没有time step,所以你dt=T/N以下的我看不太懂你的意思。
u_n在谱方法里面是指第n个基的系数么？
这个要求趋近于零，不然级数的和不存在了。
怎么会有||u_N|| <= c exp(a T)这个式子呢？
例如举个最简单的例子，
dx/dt=x,x(0)=1
用1,t,t^2,t^3,...做基函数。
例如开始我确定，要求到N个系数这个精度。
通过比较得到N-1个线性方程组，再加上一个满足边界条件的方程，
这样得到了N个方程，N个未知数。然后求解。
这样就得到了Exp(t)的N项Taylor级数展开。
这个过程中，稳定性是什么意思呢？ | | | d*********g
发帖数: 49 | 11 这是另外一种数值方法？什么是
chebyshev级数？有什么reference吗？
【在 t**********r 的大作中提到】
: 谱方法没有time step,所以你dt=T/N以下的我看不太懂你的意思。
: u_n在谱方法里面是指第n个基的系数么？
: 这个要求趋近于零，不然级数的和不存在了。
: 怎么会有||u_N|| <= c exp(a T)这个式子呢？
: 例如举个最简单的例子，
: dx/dt=x,x(0)=1
: 用1,t,t^2,t^3,...做基函数。
: 例如开始我确定，要求到N个系数这个精度。
: 通过比较得到N-1个线性方程组，再加上一个满足边界条件的方程，
: 这样得到了N个方程，N个未知数。然后求解。
| d*********g
发帖数: 49 | 12 U_t(t,x,y)+sup_{|p|
F(U(t,x,y))=0
U(T,x,y)=u(x)
本来是cauchy 问题，
但是如果加个boundary condition, 能算出来，我也满足了。
【在 c*******e 的大作中提到】
: 你把问题贴出来吧,包括bc,
: finite difference和finite element两回事
: fe里面的数学背景要高一个数量级
| w**d
发帖数: 2334 | 13 depends on your problem. Spectral methods are global methods, which
need the solution to be smooth.Otherwise, there will be oscillations
(Gibbs phenomenon).
【在 d*********g 的大作中提到】
: U_t(t,x,y)+sup_{|p|: F(U(t,x,y))=0
: U(T,x,y)=u(x)
: 本来是cauchy 问题，
: 但是如果加个boundary condition, 能算出来，我也满足了。
| d*********g
发帖数: 49 | 14 nod, nod, 长见识，长见识。。。。我索性再偏一下，如果用mathematica, matlab之类
的
自己的toolbox,解那样的非线性方程，不管用什么方法，第一能不能解，（如果稳定性收
敛
性都有，虽然我证不出来），第二要算多久，比如状态变量是两维的，不带什么超级计算
机。
【在 t**********r 的大作中提到】
: 谱方法没有time step,所以你dt=T/N以下的我看不太懂你的意思。
: u_n在谱方法里面是指第n个基的系数么？
: 这个要求趋近于零，不然级数的和不存在了。
: 怎么会有||u_N|| <= c exp(a T)这个式子呢？
: 例如举个最简单的例子，
: dx/dt=x,x(0)=1
: 用1,t,t^2,t^3,...做基函数。
: 例如开始我确定，要求到N个系数这个精度。
: 通过比较得到N-1个线性方程组，再加上一个满足边界条件的方程，
: 这样得到了N个方程，N个未知数。然后求解。
| t**********r
发帖数: 256 | 15 shooting method是什么？不懂啊。
I would write this ode into a set of first order odes'
and use the shooting method combined with the Euler's method
【在 c*******e 的大作中提到】
: 你把问题贴出来吧,包括bc,
: finite difference和finite element两回事
: fe里面的数学背景要高一个数量级
| w**d
发帖数: 2334 | 16 It is very tricky for Chebyshev collocation, because the weight function.
You can check David Gottlieb's work. He did some work about 20 years ago.
It surely has the issue of stability, no matter it is onestep or multistep.
For finite difference, if the problem is periodic, you can use Von Neumann
stability analysis; if the problem is IBVP, things become much more
complicated, but we still have general ways to check the stability of the
boundary conditions (which is essentially due to the fact | t**********r
发帖数: 256 | 17 假定U是t,x,y的Chebyshev级数或者别的级数。
然后带进去两边比较系数。
然后用数值方法求解关于系数的非线性代数方程。
【在 d*********g 的大作中提到】
: U_t(t,x,y)+sup_{|p|: F(U(t,x,y))=0
: U(T,x,y)=u(x)
: 本来是cauchy 问题，
: 但是如果加个boundary condition, 能算出来，我也满足了。
| c****n
发帖数: 2031 | 18 It surely has the issue of stability, no matter it is onestep or multistep.
For finite difference, if the problem is periodic, you can use Von Neumann
stability analysis; if the problem is IBVP, things become much more
complicated, but we still have general ways to check the stability of the
boundary conditions (which is essentially due to the fact that the
corresponding discrete linear operator is diagonalizable, e.g. by Fourier-
Laplace transform). What I'd like to know is that for Chebychev s
【在 t**********r 的大作中提到】
: 假定U是t,x,y的Chebyshev级数或者别的级数。
: 然后带进去两边比较系数。
: 然后用数值方法求解关于系数的非线性代数方程。
| c****n
发帖数: 2031 | 19 trapezoidal
【在 t**********r 的大作中提到】
: 假定U是t,x,y的Chebyshev级数或者别的级数。
: 然后带进去两边比较系数。
: 然后用数值方法求解关于系数的非线性代数方程。
| g******a
发帖数: 69 | 20 Then, I am further convinced that sth is long then sth is short. :)
【在 d*********g 的大作中提到】
: nod, nod, 长见识，长见识。。。。我索性再偏一下，如果用mathematica, matlab之类
: 的
: 自己的toolbox,解那样的非线性方程，不管用什么方法，第一能不能解，（如果稳定性收
: 敛
: 性都有，虽然我证不出来），第二要算多久，比如状态变量是两维的，不带什么超级计算
: 机。
| | | t**********r
发帖数: 256 | 21 这不就是分段折线插值么。
【在 c****n 的大作中提到】
: trapezoidal
| t**********r
发帖数: 256 | | g******a
发帖数: 69 | 23 Time is also discretized: u_n=u( ,n*dt)
There's a theorem in the theory of FDM: stability is
equivalent to convergence. If I remember correctly.
So it seems to me spectral method only works for linear equations?
For nonlinear equation (with \sup{..}) how to compare coefficients then?
【在 t**********r 的大作中提到】
: 谱方法没有time step,所以你dt=T/N以下的我看不太懂你的意思。
: u_n在谱方法里面是指第n个基的系数么？
: 这个要求趋近于零，不然级数的和不存在了。
: 怎么会有||u_N|| <= c exp(a T)这个式子呢？
: 例如举个最简单的例子，
: dx/dt=x,x(0)=1
: 用1,t,t^2,t^3,...做基函数。
: 例如开始我确定，要求到N个系数这个精度。
: 通过比较得到N-1个线性方程组，再加上一个满足边界条件的方程，
: 这样得到了N个方程，N个未知数。然后求解。
| c*******e
发帖数: 8624 | 24 I would write this ode into a set of first order odes'
and use the shooting method combined with the Euler's method
【在 t**********r 的大作中提到】
: 都说得很好，我长了不少知识阿。
| t**********r
发帖数: 256 | 25 http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/8all.pdf
stability一章 not yet written......
【在 w**d 的大作中提到】
: It is very tricky for Chebyshev collocation, because the weight function.
: You can check David Gottlieb's work. He did some work about 20 years ago.
:
: It surely has the issue of stability, no matter it is onestep or multistep.
: For finite difference, if the problem is periodic, you can use Von Neumann
: stability analysis; if the problem is IBVP, things become much more
: complicated, but we still have general ways to check the stability of the
: boundary conditions (which is essentially due to the fact
| c*******e
发帖数: 8624 | 26 ode45 solves for the values at the nodes
you can use linear interpolation if you wish
in general, if the solution is sufficiently nice
the error should go to 0 as you increase the
number of nodes
【在 t**********r 的大作中提到】
: http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/8all.pdf
: stability一章 not yet written......
| g******a
发帖数: 69 | 27 Matlab draws graph from a set of data, not a continuous function.
you can use sum to approximate integral anyway.
I just want to say usually people don't treat the solutions of
FD equations as piecewise linear functions. But in FEM, they do.
【在 t**********r 的大作中提到】
: http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/8all.pdf
: stability一章 not yet written......
| t**********r
发帖数: 256 | 28 mathematica不能实用，太慢了。
可以做概念验证方法对不对。
类
收
算
【在 d*********g 的大作中提到】
: nod, nod, 长见识，长见识。。。。我索性再偏一下，如果用mathematica, matlab之类
: 的
: 自己的toolbox,解那样的非线性方程，不管用什么方法，第一能不能解，（如果稳定性收
: 敛
: 性都有，虽然我证不出来），第二要算多久，比如状态变量是两维的，不带什么超级计算
: 机。
| c*******e
发帖数: 8624 | 29 你把问题贴出来吧,包括bc,
finite difference和finite element两回事
fe里面的数学背景要高一个数量级
【在 d*********g 的大作中提到】
: 因为我是做viscosity solution, 要求这个scheme, 比如u(t,x)=(Tu)(t-delta,dot)
: 这个operator T是单调的。好像有限元里面有什么proper还是什么性质的讲法，
: 有点类似，不好意思，我一知半解地说。
| g******a
发帖数: 69 | 30 This is not the point.
The problem is convergence or stability...
Also, finite difference solution is linear in each intervals???
You must mess it up with FEM.
【在 t**********r 的大作中提到】
: mathematica不能实用，太慢了。
: 可以做概念验证方法对不对。
:
: 类
: 收
: 算
| | | t**********r
发帖数: 256 | 31
我怀疑谱方法的稳定性目前没有确切的公理化定义。
至少没有差分法课本上那么全的稳定性的定义。
因为不同的基函数差的还是挺远的。
【在 c****n 的大作中提到】
: It surely has the issue of stability, no matter it is onestep or multistep.
: For finite difference, if the problem is periodic, you can use Von Neumann
: stability analysis; if the problem is IBVP, things become much more
: complicated, but we still have general ways to check the stability of the
: boundary conditions (which is essentially due to the fact that the
: corresponding discrete linear operator is diagonalizable, e.g. by Fourier-
: Laplace transform). What I'd like to know is that for Chebychev s
| c*******e
发帖数: 8624 | 32 I would write this ode into a set of first order odes'
and use the shooting method combined with the Euler's method
【在 t**********r 的大作中提到】
:
: 我怀疑谱方法的稳定性目前没有确切的公理化定义。
: 至少没有差分法课本上那么全的稳定性的定义。
: 因为不同的基函数差的还是挺远的。
| t**********r
发帖数: 256 | 33 书上说原则上来说，混合法速度最快。
先划分小点的区间，然后在区间上用谱方法。
谱方法得到的照样是数值解，差分法得到的是分段线性插直函数
谱方法得到分段多项式。
【在 d*********g 的大作中提到】
: nod, nod, 长见识，长见识。。。。我索性再偏一下，如果用mathematica, matlab之类
: 的
: 自己的toolbox,解那样的非线性方程，不管用什么方法，第一能不能解，（如果稳定性收
: 敛
: 性都有，虽然我证不出来），第二要算多久，比如状态变量是两维的，不带什么超级计算
: 机。
| t**********r
发帖数: 256 | 34 不过matlab的ODE45之类的东西，
给出来的插值函数是默认线性插值的吧？
mathematica好像也是。
很多时候，我不仅需要某点的值，我需要所有点的值。
例如求解了ODE之后，需要画图plot(x(t),t)
或者求x(t)^2的积分阿。
【在 g******a 的大作中提到】
: This is not the point.
: The problem is convergence or stability...
: Also, finite difference solution is linear in each intervals???
: You must mess it up with FEM.
| c*******e
发帖数: 8624 | 35 ode45 solves for the values at the nodes
you can use linear interpolation if you wish
in general, if the solution is sufficiently nice
the error should go to 0 as you increase the
number of nodes
【在 t**********r 的大作中提到】
: 不过matlab的ODE45之类的东西，
: 给出来的插值函数是默认线性插值的吧？
: mathematica好像也是。
: 很多时候，我不仅需要某点的值，我需要所有点的值。
: 例如求解了ODE之后，需要画图plot(x(t),t)
: 或者求x(t)^2的积分阿。
| sc
发帖数: 122 | | t**********r
发帖数: 256 | 37 谱方法更加archaic, Newton就开始用了.
【在 w**d 的大作中提到】
: It is very tricky for Chebyshev collocation, because the weight function.
: You can check David Gottlieb's work. He did some work about 20 years ago.
:
: It surely has the issue of stability, no matter it is onestep or multistep.
: For finite difference, if the problem is periodic, you can use Von Neumann
: stability analysis; if the problem is IBVP, things become much more
: complicated, but we still have general ways to check the stability of the
: boundary conditions (which is essentially due to the fact
| d*********g
发帖数: 49 | 38 Politically correct!
Although my problems remain unsolved, I benefit a lot from the previous disc
ussion. My research area is not on whatever numerical analysis, but it seems
from here,as well as my other resourses, mathematicians are more interested in
spectrul methods than FD nowadays, though FE still a big challenge. I am
certainly no authority on commenting on this,just my observation. For me, I
just like to go for FD because it's so intuitive.
You reminds me of another matter, or another of | c******m
发帖数: 599 | 39 别别
spectral 从来没有和FD/FE一样流行过
在数学圈也这样
没你想的那么随波逐流
in
【在 d*********g 的大作中提到】
: Politically correct!
: Although my problems remain unsolved, I benefit a lot from the previous disc
: ussion. My research area is not on whatever numerical analysis, but it seems
: from here,as well as my other resourses, mathematicians are more interested in
: spectrul methods than FD nowadays, though FE still a big challenge. I am
: certainly no authority on commenting on this,just my observation. For me, I
: just like to go for FD because it's so intuitive.
: You reminds me of another matter, or another of
| t**********r
发帖数: 256 | 40 nod
这个方法的缺点是最后要处理非线性代数方程组，
而有些非线性代数方程没什么好的办法求解。
或者用牛顿法把问题转化成一系列的线性PDE然后解一系列的线性代数方程
收敛性很难预计。
【在 w**d 的大作中提到】
: It is very tricky for Chebyshev collocation, because the weight function.
: You can check David Gottlieb's work. He did some work about 20 years ago.
:
: It surely has the issue of stability, no matter it is onestep or multistep.
: For finite difference, if the problem is periodic, you can use Von Neumann
: stability analysis; if the problem is IBVP, things become much more
: complicated, but we still have general ways to check the stability of the
: boundary conditions (which is essentially due to the fact
| | | c****n
发帖数: 2031 | 41 You mean you solve the semi-discrete equation analytically and then
truncate the summation to obtain a numerical solution? I think that's
only one of the spectral methods.
Anyway, I think we are far away from the original problem right now, hehe.
【在 t**********r 的大作中提到】
: nod
: 这个方法的缺点是最后要处理非线性代数方程组，
: 而有些非线性代数方程没什么好的办法求解。
: 或者用牛顿法把问题转化成一系列的线性PDE然后解一系列的线性代数方程
: 收敛性很难预计。
| c****n
发帖数: 2031 | 42 Different people may have different opinions. Besides, doing research is not
all about chasing behind fashion.
【在 d*********g 的大作中提到】
: Politically correct!
: Although my problems remain unsolved, I benefit a lot from the previous disc
: ussion. My research area is not on whatever numerical analysis, but it seems
: from here,as well as my other resourses, mathematicians are more interested in
: spectrul methods than FD nowadays, though FE still a big challenge. I am
: certainly no authority on commenting on this,just my observation. For me, I
: just like to go for FD because it's so intuitive.
: You reminds me of another matter, or another of
| t**********r
发帖数: 256 | 43 Nod.
If the solution is continuous function, we can use Chebyshev polynomials
to approximate the solution to infinite precision.
The drawback is, we have to deal with a nonlinear algebra
equations that may be very senstive to numerical error.
【在 w**d 的大作中提到】
: It is very tricky for Chebyshev collocation, because the weight function.
: You can check David Gottlieb's work. He did some work about 20 years ago.
:
: It surely has the issue of stability, no matter it is onestep or multistep.
: For finite difference, if the problem is periodic, you can use Von Neumann
: stability analysis; if the problem is IBVP, things become much more
: complicated, but we still have general ways to check the stability of the
: boundary conditions (which is essentially due to the fact
| w**d
发帖数: 2334 | 44 depends on your problem. Spectral methods are global methods, which
need the solution to be smooth.Otherwise, there will be oscillations
(Gibbs phenomenon).
【在 d*********g 的大作中提到】
: Politically correct!
: Although my problems remain unsolved, I benefit a lot from the previous disc
: ussion. My research area is not on whatever numerical analysis, but it seems
: from here,as well as my other resourses, mathematicians are more interested in
: spectrul methods than FD nowadays, though FE still a big challenge. I am
: certainly no authority on commenting on this,just my observation. For me, I
: just like to go for FD because it's so intuitive.
: You reminds me of another matter, or another of
| c****n
发帖数: 2031 | 45 It surely has the issue of stability, no matter it is onestep or multistep.
For finite difference, if the problem is periodic, you can use Von Neumann
stability analysis; if the problem is IBVP, things become much more
complicated, but we still have general ways to check the stability of the
boundary conditions (which is essentially due to the fact that the
corresponding discrete linear operator is diagonalizable, e.g. by Fourier-
Laplace transform). What I'd like to know is that for Chebychev s
【在 t**********r 的大作中提到】
: Nod.
: If the solution is continuous function, we can use Chebyshev polynomials
: to approximate the solution to infinite precision.
: The drawback is, we have to deal with a nonlinear algebra
: equations that may be very senstive to numerical error.
| w**d
发帖数: 2334 | 46 So, what is not archaic?
【在 d*********g 的大作中提到】
: Politically correct!
: Although my problems remain unsolved, I benefit a lot from the previous disc
: ussion. My research area is not on whatever numerical analysis, but it seems
: from here,as well as my other resourses, mathematicians are more interested in
: spectrul methods than FD nowadays, though FE still a big challenge. I am
: certainly no authority on commenting on this,just my observation. For me, I
: just like to go for FD because it's so intuitive.
: You reminds me of another matter, or another of
| c****n
发帖数: 2031 | 47 thanks man
【在 w**d 的大作中提到】
: So, what is not archaic?
| w**d
发帖数: 2334 | 48 It will be very beneficial to read some book.
For time dependent problem, the definition of stability
should be the same for whatever schemes. It basically means
the energy is bounded.
Assume the final time is T, the time step dt = T/N.
Let u_n be numerical solu at t=t_n. For finite difference/element,
u_n is mostly piecewise polynomials, not necessary linear.
For spectral methods, u_n will be a global polynomial.
Just like for u, we can define some norm(energy) for u_n. Then
the stability means | t**********r
发帖数: 256 | 49 假定U是t,x,y的Chebyshev级数或者别的级数。
然后带进去两边比较系数。
然后用数值方法求解关于系数的非线性代数方程。
【在 d*********g 的大作中提到】
: U_t(t,x,y)+sup_{|p|: F(U(t,x,y))=0
: U(T,x,y)=u(x)
: 本来是cauchy 问题，
: 但是如果加个boundary condition, 能算出来，我也满足了。
| t**********r
发帖数: 256 | 50 这不就是分段折线插值么。
【在 c****n 的大作中提到】
: trapezoidal
| | | t**********r
发帖数: 256 | 51 都说得很好，我长了不少知识阿。
【在 sc 的大作中提到】
: 各位老大，偏题了..
| g******a
发帖数: 69 | 52
right up to here.
Usually, people don't do that. If you are interested in the value of f
at certain point, just choose that point as a nodel.
【在 t**********r 的大作中提到】
: 都说得很好，我长了不少知识阿。
| t**********r
发帖数: 256 | 53 其实Chebyshev方法连最简单的一阶ODE都保证不了能求解,
如果函数的Chebyshev系数不能精确给出的话。
这是2003年一本最新的书上说的。
好在很多问题建模就是用特定基的形式给出来的。
【在 w**d 的大作中提到】
: It is very tricky for Chebyshev collocation, because the weight function.
: You can check David Gottlieb's work. He did some work about 20 years ago.
:
: It surely has the issue of stability, no matter it is onestep or multistep.
: For finite difference, if the problem is periodic, you can use Von Neumann
: stability analysis; if the problem is IBVP, things become much more
: complicated, but we still have general ways to check the stability of the
: boundary conditions (which is essentially due to the fact
| c****n
发帖数: 2031 | 54 Hi, could you recommend some paper of Dr. Trefethen concerning stability?
Thanks!
【在 w**d 的大作中提到】
: It will be very beneficial to read some book.
: For time dependent problem, the definition of stability
: should be the same for whatever schemes. It basically means
: the energy is bounded.
: Assume the final time is T, the time step dt = T/N.
: Let u_n be numerical solu at t=t_n. For finite difference/element,
: u_n is mostly piecewise polynomials, not necessary linear.
: For spectral methods, u_n will be a global polynomial.
: Just like for u, we can define some norm(energy) for u_n. Then
: the stability means
| c****n
发帖数: 2031 | 55
~~~~~~~~~Not really, buddy
【在 d*********g 的大作中提到】
: Politically correct!
: Although my problems remain unsolved, I benefit a lot from the previous disc
: ussion. My research area is not on whatever numerical analysis, but it seems
: from here,as well as my other resourses, mathematicians are more interested in
: spectrul methods than FD nowadays, though FE still a big challenge. I am
: certainly no authority on commenting on this,just my observation. For me, I
: just like to go for FD because it's so intuitive.
: You reminds me of another matter, or another of
| d*********g
发帖数: 49 | 56 Thank you, I will take a look at that.
My question is how to properly discretize this equation, so that
the scheme I get has certain monotonicity.
【在 w**a 的大作中提到】
: i didn't fully understand your problem,
: is your problem same to the 'upwind f.d. scheme' in viscosity solution
: or 'entropy 'condition?
| w**d
发帖数: 2334 | 57 that is spectral method, i believe.
【在 d*********g 的大作中提到】
: 这是另外一种数值方法？什么是
: chebyshev级数？有什么reference吗？
| c******m
发帖数: 599 | 58 ft
被无数工程界人士bs死
【在 d*********g 的大作中提到】
: Thank you, I will take a look at that.
: My question is how to properly discretize this equation, so that
: the scheme I get has certain monotonicity.
| t**********r
发帖数: 256 | 59 http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/8all.pdf
stability一章 not yet written......
【在 w**d 的大作中提到】
: that is spectral method, i believe.
| g******a
发帖数: 69 | 60 exactly.
【在 c*******e 的大作中提到】
: ode45 solves for the values at the nodes
: you can use linear interpolation if you wish
: in general, if the solution is sufficiently nice
: the error should go to 0 as you increase the
: number of nodes
| | | c****n
发帖数: 2031 | 61 Hehe, I asked about stability (for linear PDE). Without it, even the scheme
has very high order of accuracy, it may still diverge.
【在 t**********r 的大作中提到】
: http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/8all.pdf
: stability一章 not yet written......
| w**d
发帖数: 2334 | 62 It is very tricky for Chebyshev collocation, because the weight function.
You can check David Gottlieb's work. He did some work about 20 years ago.
It surely has the issue of stability, no matter it is onestep or multistep.
For finite difference, if the problem is periodic, you can use Von Neumann
stability analysis; if the problem is IBVP, things become much more
complicated, but we still have general ways to check the stability of the
boundary conditions (which is essentially due to the fact
【在 c****n 的大作中提到】
: It surely has the issue of stability, no matter it is onestep or multistep.
: For finite difference, if the problem is periodic, you can use Von Neumann
: stability analysis; if the problem is IBVP, things become much more
: complicated, but we still have general ways to check the stability of the
: boundary conditions (which is essentially due to the fact that the
: corresponding discrete linear operator is diagonalizable, e.g. by Fourier-
: Laplace transform). What I'd like to know is that for Chebychev s
| d*********g
发帖数: 49 | 63 这是另外一种数值方法？什么是
chebyshev级数？有什么reference吗？
【在 t**********r 的大作中提到】
: 假定U是t,x,y的Chebyshev级数或者别的级数。
: 然后带进去两边比较系数。
: 然后用数值方法求解关于系数的非线性代数方程。
| c******m
发帖数: 599 | 64 别别
spectral 从来没有和FD/FE一样流行过
在数学圈也这样
没你想的那么随波逐流
in
【在 d*********g 的大作中提到】
: Politically correct!
: Although my problems remain unsolved, I benefit a lot from the previous disc
: ussion. My research area is not on whatever numerical analysis, but it seems
: from here,as well as my other resourses, mathematicians are more interested in
: spectrul methods than FD nowadays, though FE still a big challenge. I am
: certainly no authority on commenting on this,just my observation. For me, I
: just like to go for FD because it's so intuitive.
: You reminds me of another matter, or another of
|
|
