finite element method for the Shallow Water Equations help

alvesker · December 4, 2011, 22:28

Hello,

I try to solve the Shallow Water Equations using finite element methods. Can anyone explain me how to treat nonlinear terms?

The problem is that we can write linear terms as matrix-vector multiplication where matrix is independent on time. For example for the height field h we have

$h=\sum_{i=1,N}c_i^hB_i$

after getting the Galerkin equation by multiplying on test function and integrating over domain we will get the term like

$\sum_{i=1,N}c_i^h\int B_iB_jdx$

or in the matrix form

$(\int B_iB_jdx)_{i,j=1,N}(c_i^h)_{i=1,N}$

Hence we compute the matrix only once and then multiply by updated coefficient vector

But how to deal with the terms like $uh$? where u is the velocity

$uh=\sum_{i,k=1,N}c_i^uc_k^hB_iB_k$

after getting the Galerkin equation this will be like

$\sum_{i,k=1,N}c_i^uc_k^h\int B_iB_jB_kdx$

How these can be represented in the matrix form in order not to compute all these integrals on each timestep?

seyedashraf · December 5, 2011, 03:31

Dear alvesker,

you don't have to write the FEM approximated form of each parameter

is more accurate, though, I would suggest writing the the mean value of those parameters to simplify the weak form.

about the integration on each time step
it depends on the grid form you are using to split the domain.

alvesker · December 5, 2011, 05:48

Dear seyedashraf,

Thanks for the quick reply.

Can you explain what do you understand by the mean value?
is it like the following?

$hu = \sum_{i=1,N}c_i^hc_i^u B_i$

and how write it to the terms like

$u\nabla h$

where u, h are the velocity component and height field

$u = \sum_{i=1,N}c_i^uB_i,\ h=\sum_{i=1,N}c_i^hB_i$

So if apply the FEM we have

$u\nabla h=\sum_{i=1,N}c_i^uB_i\sum_{i=1,N}c_i^h\nabla B_i=\sum_{i,j=1,N}c_i^uc_j^hB_i\nabla B_j$

what is the average in the case when in the multiplication one of unknows one contain gradient (we have these in the advection formulation)

alvesker · December 5, 2011, 06:01

the grid form - I use triangular grid

December 4, 2011, 22:28	finite element method for the Shallow Water Equations help	#1
alvesker New Member John Smith Join Date: Dec 2011 Posts: 4 Rep Power: 14	Hello, I try to solve the Shallow Water Equations using finite element methods. Can anyone explain me how to treat nonlinear terms? The problem is that we can write linear terms as matrix-vector multiplication where matrix is independent on time. For example for the height field h we have $h=\sum_{i=1,N}c_i^hB_i$ after getting the Galerkin equation by multiplying on test function and integrating over domain we will get the term like $\sum_{i=1,N}c_i^h\int B_iB_jdx$ or in the matrix form $(\int B_iB_jdx)_{i,j=1,N}(c_i^h)_{i=1,N}$ Hence we compute the matrix only once and then multiply by updated coefficient vector But how to deal with the terms like $uh$? where u is the velocity $uh=\sum_{i,k=1,N}c_i^uc_k^hB_iB_k$ after getting the Galerkin equation this will be like $\sum_{i,k=1,N}c_i^uc_k^h\int B_iB_jB_kdx$ How these can be represented in the matrix form in order not to compute all these integrals on each timestep? Last edited by alvesker; December 4, 2011 at 22:54. Reason: typo

December 5, 2011, 03:31		#2
seyedashraf Member Omid Seyedashraf Join Date: May 2010 Posts: 49 Rep Power: 16	Dear alvesker, you don't have to write the FEM approximated form of each parameter is more accurate, though, I would suggest writing the the mean value of those parameters to simplify the weak form. about the integration on each time step it depends on the grid form you are using to split the domain. Last edited by seyedashraf; December 5, 2011 at 05:52.

December 5, 2011, 05:48		#3
alvesker New Member John Smith Join Date: Dec 2011 Posts: 4 Rep Power: 14	Dear seyedashraf, Thanks for the quick reply. Can you explain what do you understand by the mean value? is it like the following? $hu = \sum_{i=1,N}c_i^hc_i^u B_i$ and how write it to the terms like $u\nabla h$ where u, h are the velocity component and height field $u = \sum_{i=1,N}c_i^uB_i,\ h=\sum_{i=1,N}c_i^hB_i$ So if apply the FEM we have $u\nabla h=\sum_{i=1,N}c_i^uB_i\sum_{i=1,N}c_i^h\nabla B_i=\sum_{i,j=1,N}c_i^uc_j^hB_i\nabla B_j$ what is the average in the case when in the multiplication one of unknows one contain gradient (we have these in the advection formulation) Last edited by alvesker; December 6, 2011 at 00:20. Reason: add

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December 5, 2011, 06:01		#4
alvesker New Member John Smith Join Date: Dec 2011 Posts: 4 Rep Power: 14	the grid form - I use triangular grid