Solving nonlinear equation resulting from finite element method
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Using the finite element method (for a uniform mesh in the spatial domain) I have the system with initial conditions $u_j(0)=cos(x_j)$ for $j=1,dots, N$
$$frac{dvec{u}}{dt}=Avec{u}+Bvec{c},$$
where $A$ and $B$ are coefficient matrices and $vec{c}$ takes the form
$$vec{c}=begin{bmatrix}displaystyleint_0^1left(vec{u}cdotbeta(x)right)^2beta_j'(x)dxend{bmatrix}_{1leq jleq N}$$
and $u_j=u_j(t)approx u(x_j,t)$ is the semi-approximation of the solution at time $t$ on the mesh node $x_j$ for $j=1,dots, N$ and $beta_j$ is a linear basis function. My book says that this system can be solved with a "standard ODE solver", but provides no other insight besides that. My question is how does one go about solving such a system? Is there a particular MATLAB command that is used to solve this system? After searching online I haven't been able to find something conrete. Any discussion or insight would be greatly appreciated.
numerical-methods matlab nonlinear-system finite-element-method
asked Nov 23 at 21:31
user23793
625721
add a comment |
up vote
1
down vote
favorite
Using the finite element method (for a uniform mesh in the spatial domain) I have the system with initial conditions $u_j(0)=cos(x_j)$ for $j=1,dots, N$
$$frac{dvec{u}}{dt}=Avec{u}+Bvec{c},$$
where $A$ and $B$ are coefficient matrices and $vec{c}$ takes the form
$$vec{c}=begin{bmatrix}displaystyleint_0^1left(vec{u}cdotbeta(x)right)^2beta_j'(x)dxend{bmatrix}_{1leq jleq N}$$
and $u_j=u_j(t)approx u(x_j,t)$ is the semi-approximation of the solution at time $t$ on the mesh node $x_j$ for $j=1,dots, N$ and $beta_j$ is a linear basis function. My book says that this system can be solved with a "standard ODE solver", but provides no other insight besides that. My question is how does one go about solving such a system? Is there a particular MATLAB command that is used to solve this system? After searching online I haven't been able to find something conrete. Any discussion or insight would be greatly appreciated.
numerical-methods matlab nonlinear-system finite-element-method
asked Nov 23 at 21:31
user23793
625721
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Using the finite element method (for a uniform mesh in the spatial domain) I have the system with initial conditions $u_j(0)=cos(x_j)$ for $j=1,dots, N$
$$frac{dvec{u}}{dt}=Avec{u}+Bvec{c},$$
where $A$ and $B$ are coefficient matrices and $vec{c}$ takes the form
$$vec{c}=begin{bmatrix}displaystyleint_0^1left(vec{u}cdotbeta(x)right)^2beta_j'(x)dxend{bmatrix}_{1leq jleq N}$$
and $u_j=u_j(t)approx u(x_j,t)$ is the semi-approximation of the solution at time $t$ on the mesh node $x_j$ for $j=1,dots, N$ and $beta_j$ is a linear basis function. My book says that this system can be solved with a "standard ODE solver", but provides no other insight besides that. My question is how does one go about solving such a system? Is there a particular MATLAB command that is used to solve this system? After searching online I haven't been able to find something conrete. Any discussion or insight would be greatly appreciated.
numerical-methods matlab nonlinear-system finite-element-method
asked Nov 23 at 21:31
user23793
625721
Using the finite element method (for a uniform mesh in the spatial domain) I have the system with initial conditions $u_j(0)=cos(x_j)$ for $j=1,dots, N$
$$frac{dvec{u}}{dt}=Avec{u}+Bvec{c},$$
where $A$ and $B$ are coefficient matrices and $vec{c}$ takes the form
$$vec{c}=begin{bmatrix}displaystyleint_0^1left(vec{u}cdotbeta(x)right)^2beta_j'(x)dxend{bmatrix}_{1leq jleq N}$$
and $u_j=u_j(t)approx u(x_j,t)$ is the semi-approximation of the solution at time $t$ on the mesh node $x_j$ for $j=1,dots, N$ and $beta_j$ is a linear basis function. My book says that this system can be solved with a "standard ODE solver", but provides no other insight besides that. My question is how does one go about solving such a system? Is there a particular MATLAB command that is used to solve this system? After searching online I haven't been able to find something conrete. Any discussion or insight would be greatly appreciated.
numerical-methods matlab nonlinear-system finite-element-method
numerical-methods matlab nonlinear-system finite-element-method
asked Nov 23 at 21:31
user23793
625721
asked Nov 23 at 21:31
user23793
625721
edited Nov 23 at 21:52
asked Nov 23 at 21:31
user23793
625721
asked Nov 23 at 21:31
user23793
625721
asked Nov 23 at 21:31
user23793
625721
625721
add a comment |
add a comment |
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