"Solve the system with initial value $(1,0,0)$ at $0$ with the time range$0 \\leq\n",
"t \\leq 3$. Use at least two different solvers and compare their results. Try to understand heuristicly why different solvers behave sometimes very differently."
],
"cell_type": "markdown"
},
{
"metadata": {},
"source": [
"##problem 2\n",
"\n",
"Implement an explicit Euler method to solve ODE's. The explicit euler method has the parameters $c_i = 0$, $a_{ij} = 0$ and $b_1 = 0$ which then yields the update formula\n",
"\n",
"$$y_{n + 1} = y_n + h f ( t_n , y_ n )$$\n",
"\n",
"Choose a value h for the size of every step and set $t_n = t_0 + n h$ . \n"
],
"cell_type": "markdown"
},
{
"metadata": {},
"source": [
"## problem 3\n",
"\n",
"*stiff* ODE's are somewhat *ill-conditioned* in the sense, that many algorithms for solving this type of ODE are then unstable. Consider the initial value problem: \n",
"\n",
"$$y^\\prime ( t ) = - 15 y ( t ) , t \\ge 0 , y ( 0 ) = 1$$\n",
"\n",
"and solve it with the `odeSolve` and the explicit euler. Look especially at $h= 1/4$ and $h = 1/8$ for the euler method and look what is happening.\n",
Solve the system with initial value $(1,0,0)$ at $0$ with the time range$0 \leq
t \leq 3$. Use at least two different solvers and compare their results. Try to understand heuristicly why different solvers behave sometimes very differently.
%% Cell type:markdown id: tags:
##problem 2
Implement an explicit Euler method to solve ODE's. The explicit euler method has the parameters $c_i = 0$, $a_{ij} = 0$ and $b_1 = 0$ which then yields the update formula
$$y_{n + 1} = y_n + h f ( t_n , y_ n )$$
Choose a value h for the size of every step and set $t_n = t_0 + n h$ .
%% Cell type:markdown id: tags:
## problem 3
*stiff* ODE's are somewhat *ill-conditioned* in the sense, that many algorithms for solving this type of ODE are then unstable. Consider the initial value problem:
$$y^\prime ( t ) = - 15 y ( t ) , t \ge 0 , y ( 0 ) = 1$$
and solve it with the `odeSolve` and the explicit euler. Look especially at $h= 1/4$ and $h = 1/8$ for the euler method and look what is happening.
"Solve the system with initial value $(1,0,0)$ at $0$ with the time range$0 \\leq\n",
"t \\leq 3$. Use at least two different solvers and compare their results."
"t \\leq 3$. Use at least two different solvers and compare their results. Try to understand heuristicly why different solvers behave sometimes very differently."
]
},
{
...
...
@@ -131,11 +131,11 @@
"source": [
"## problem 3\n",
"\n",
"*stiff* ODE's: Consider the initial value problem: \n",
"*stiff* ODE's are somewhat *ill-conditioned* in the sense, that many algorithms for solving this type of ODE are then unstable. Consider the initial value problem: \n",
"\n",
"$$y^\\prime ( t ) = - 15 y ( t ) , t \\ge 0 , y ( 0 ) = 1$$\n",
"\n",
"and solve it with the odesolve\n",
"and solve it with the `odeSolve` and the explicit euler. Look especially at $h= 1/4$ and $h = 1/8$ for the euler method and look what is happening.\n",
Solve the system with initial value $(1,0,0)$ at $0$ with the time range$0 \leq
t \leq 3$. Use at least two different solvers and compare their results.
t \leq 3$. Use at least two different solvers and compare their results. Try to understand heuristicly why different solvers behave sometimes very differently.
*stiff* ODE's: Consider the initial value problem:
*stiff* ODE's are somewhat *ill-conditioned* in the sense, that many algorithms for solving this type of ODE are then unstable. Consider the initial value problem:
$$y^\prime ( t ) = - 15 y ( t ) , t \ge 0 , y ( 0 ) = 1$$
and solve it with the odesolve
and solve it with the `odeSolve` and the explicit euler. Look especially at $h= 1/4$ and $h = 1/8$ for the euler method and look what is happening.