Splitting methods#
Some methods to solve a problem of the form:
with the initial condition \(u(t=0)=u_0\). We note with \(\phi_{\tau}^{[f_i]}(t^n,\tilde{u}^n)\) the solution at time \(t^n+\tau\) of the subproblem \(\dot{u}=f_i(t,u)\) with the initial condition \(u(t^n)=\tilde{u}^n\).
Lie splitting method#
In Lie splitting method, the solution is computed as:
-
template<typename value_t, typename ...Methods_t>
class lie# Lie splitting method.
- Template Parameters:
value_t – type of time steps
Methods_t – list of methods to solve each sub-problem
Public Functions
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lie(std::tuple<Methods_t...> const &meths, std::array<value_t, sizeof...(Methods_t)> const &dts)#
constructor of lie from a tuple
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template<std::size_t I = 0, typename Problem_t, typename state_t>
inline void _call_inc(Problem_t &f, value_t tn, state_t &ui, value_t dt)# incremental call of each method of each subproblem
The parameter
uiis update to \(\phi_{\Delta t}^{[f_i]}(t^n,\texttt{ui})\)- Template Parameters:
I – solving step
- Parameters:
f – problem to solve
tn – current time \(t^n\)
ui – [inout] \(\texttt{ui}=\phi_{\Delta t}^{[f_1]}\circ\cdots\circ\phi_{\Delta t}^{[f_{i-1}]}(t^n,u^n)\)
dt – time step \(\Delta t\)
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template<typename Problem_t, typename state_t>
auto operator()(Problem_t &f, value_t tn, state_t const &un, value_t dt)# call operator to initiate Lie splitting recursion
- Parameters:
f – problem to solve
tn – current time \(t^n\)
un – current solution \(u^n \approx u(t^n)\)
dt – time step \(\Delta t\)
Helper functions and classes#
-
template<typename value_t, typename ...Methods_t>
auto ponio::splitting::lie::make_lie_from_tuple(std::tuple<Methods_t...> const &meths, std::array<value_t, sizeof...(Methods_t)> const &dts)# a helper factory for lie functor from a tuple of methods
- Template Parameters:
value_t – type of coefficients
Methods_t – variadic list of types of methods
- Parameters:
meths – list of methods
dts – associated time step foreach method
- Returns:
a lie object build from the tuple of methods
-
template<typename value_t, typename ...Algorithms_t>
class lie_tuple# a helper to deduce method for ponio::make_method(splitting::lie::lie_tuple<Algorithms_t…> const &, state_t const &)
This is a dummy class to select correct method to solve the problem
- Template Parameters:
value_t – type of time steps
Algorithms_t – variadic template of algorithms to solve each subproblem
Public Functions
-
inline lie_tuple(std::tuple<Algorithms_t...> &&algs, std::array<value_t, sizeof...(Algorithms_t)> &&dts)#
constructor of lie_tuple from a variadic number of algorithms
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template<typename value_t, typename ...Algorithms_t>
auto ponio::splitting::lie::make_lie_tuple(std::pair<Algorithms_t, value_t>&&... args)# a helper factory for lie_tuple from a tuple of algorithms
- Template Parameters:
value_t – type of coefficients
Algorithms_t – variadic list of types of algorithms
- Parameters:
args – variadic list of pairs of algorithm and time step
- Returns:
a lie_tuple object build from the tuple of methods
Strang splitting method#
In Strang splitting method, the solution is computed as:
-
template<typename value_t, typename ...Methods_t>
class strang : public ponio::splitting::lie::lie<value_t, Methods_t...># Strang splitting method.
- Template Parameters:
Methods_t – list of methods to solve each sub-problem
Public Functions
-
template<std::size_t I = 0, typename Problem_t, typename state_t>
inline void _call_inc(Problem_t &f, value_t tn, state_t &ui, value_t dt)# incremental call of each method of each subproblem
The parameter
uiis update to \(\phi_{^{\Delta t}/_2}^{[f_i]}(t^n,\texttt{ui})\)- Template Parameters:
I – solving step
- Parameters:
f – problem to solve
tn – current time \(t^n\)
ui – [inout] \(\texttt{ui}=\phi_{^{\Delta t}/_2}^{[f_1]}\circ\cdots\circ\phi_{^{\Delta t}/_2}^{[f_{i-1}]}(t^n,u^n)\)
dt – time step \(\Delta t\)
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template<std::size_t I = sizeof...(Methods_t) - 1, typename Problem_t, typename state_t>
inline void _call_dec(Problem_t &f, value_t tn, state_t &ui, value_t dt)# decremental call of each method of each subproblem
The parameter
uiis update to \(\phi_{^{\Delta t}/_2}^{[f_i]}(t^n,\texttt{ui})\)- Template Parameters:
I – solving step
- Parameters:
f – problem to solve
tn – current time \(t^n\)
ui – [inout] current state of the substep I \(\texttt{ui}=\phi_{^{\Delta t}/_2}^{[f_1]}\circ\cdots\circ\phi_{\Delta t}^{[f_{n}]}\circ\cdots\circ\phi_{^{\Delta t}/_2}^{[f_{i+1}]}(t^n,u^n)\)
dt – time step \(\Delta t\)
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template<typename Problem_t, typename state_t>
auto operator()(Problem_t &f, value_t tn, state_t const &un, value_t dt)# call operator to initiate Strang splitting recursion
- Parameters:
f – problem to solve
tn – current time \(t^n\)
un – current solution \(u^n \approx u(t^n)\)
dt – time step \(\Delta t\)
Helper functions and classes#
-
template<typename value_t, typename ...Methods_t>
auto ponio::splitting::strang::make_strang_from_tuple(std::tuple<Methods_t...> const &meths, std::array<value_t, sizeof...(Methods_t)> const &dts)# a helper factory for strang functor from a tuple of methods
- Template Parameters:
value_t – type of coefficients
Methods_t – variadic list of types of methods
- Parameters:
meths – list of methods
dts – associated time step foreach method
- Returns:
a strang object build from the tuple of methods
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template<typename value_t, typename ...Algorithms_t>
class strang_tuple# a helper to deduce method for ponio::make_method(splitting::strang::strang_tuple<Algorithms_t…> const &, state_t const &)
This is a dummy class to select correct method to solve the problem
- Template Parameters:
Algorithms_t – variadic template of algorithms to solve each subproblem
Public Functions
-
inline strang_tuple(std::tuple<Algorithms_t...> &&algs, std::array<value_t, sizeof...(Algorithms_t)> &&dts)#
constructor of strang_tuple from a variadic number of algorithms
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template<typename value_t, typename ...Algorithms_t>
auto ponio::splitting::strang::make_strang_tuple(std::pair<Algorithms_t, value_t>&&... args)# a helper factory for strang_tuple from a tuple of algorithms
- Template Parameters:
value_t – type of coefficients
Algorithms_t – variadic list of types of algorithms
- Parameters:
args – variadic list of pairs of algorithm and time step
- Returns:
a strang_tuple object build from the tuple of methods