is there a book with a chapter on convergence consistency and stability for some specific numerical methods?
0
$begingroup$
The methods Im interested in are :
- Euler's Explicit and Implicit method
- Lax-Wendroff's Method
- Crank-Nicolson's Method
even online lecture notes with analysis of these would be great. thanks !
ordinary-differential-equations reference-request numerical-methods
asked Jan 6 at 17:49
rapidracimrapidracim
1,5181319
$endgroup$
|
show 2 more comments
0
$begingroup$
The methods Im interested in are :
- Euler's Explicit and Implicit method
- Lax-Wendroff's Method
- Crank-Nicolson's Method
even online lecture notes with analysis of these would be great. thanks !
ordinary-differential-equations reference-request numerical-methods
asked Jan 6 at 17:49
rapidracimrapidracim
1,5181319
$endgroup$
$begingroup$
The second and third are methods for PDE. Is that the context you want to have considered in the references?
$endgroup$
– LutzL
Jan 6 at 18:13
$begingroup$
@LutzL Actually I only know them as methods for approximating solutions of ODE's
$endgroup$
– rapidracim
Jan 6 at 18:16
$begingroup$
@LutzL here's the LW scheme that I'm talking about : $h$ is the step we subdivide an interval $(a,b)$ into many $(t_i,t_{i+1})$ with $t_0 = a,,, t_{N} = b$, the eqn is $y' =f(t,y)$, $f$ is a 'nice' function then here's the algorithm : $$t_{n+1} = t_{n} + h,, 0 leq n leq N-1, ,, L_n = hf(t_n,y_n), ,, y_{n+1} = y_{n} +hf(t_n+frac{h}{2},y_n+frac{L_n}{2})$$ $y_{n+1}$ approximates $y(t_{n+1})$
$endgroup$
– rapidracim
Jan 6 at 18:24
1
$begingroup$
I know Lax-Wendroff as a method to approximate the Burgers equation and similar, Crank-Nicolson is usually introduced to solve the heat equation and similar. If you formulate it as method of lines, it reduces to a system of ODE, but that is not the primary problem statement.
$endgroup$
– LutzL
Jan 6 at 18:26
$begingroup$
That is the explicit midpoint method, or improved Euler method. It may be used in one of the variations of the LW scheme in the configuration of the time step. You then have probably the usual misconception to give the Heun, explicit trapezoidal, modified Euler method another name (because 3 names is not enough?) because the time step of CR is configured like the implicit trapezoidal method?
$endgroup$
– LutzL
Jan 6 at 18:29
|
show 2 more comments
0
0
0
$begingroup$
The methods Im interested in are :
- Euler's Explicit and Implicit method
- Lax-Wendroff's Method
- Crank-Nicolson's Method
even online lecture notes with analysis of these would be great. thanks !
ordinary-differential-equations reference-request numerical-methods
asked Jan 6 at 17:49
rapidracimrapidracim
1,5181319
$endgroup$
The methods Im interested in are :
- Euler's Explicit and Implicit method
- Lax-Wendroff's Method
- Crank-Nicolson's Method
even online lecture notes with analysis of these would be great. thanks !
ordinary-differential-equations reference-request numerical-methods
ordinary-differential-equations reference-request numerical-methods
asked Jan 6 at 17:49
rapidracimrapidracim
1,5181319
asked Jan 6 at 17:49
rapidracimrapidracim
1,5181319
edited Jan 6 at 17:55
rapidracim
asked Jan 6 at 17:49
rapidracimrapidracim
1,5181319
asked Jan 6 at 17:49
rapidracimrapidracim
1,5181319
asked Jan 6 at 17:49
rapidracimrapidracim
1,5181319
1,5181319
$begingroup$
The second and third are methods for PDE. Is that the context you want to have considered in the references?
$endgroup$
– LutzL
Jan 6 at 18:13
$begingroup$
@LutzL Actually I only know them as methods for approximating solutions of ODE's
$endgroup$
– rapidracim
Jan 6 at 18:16
$begingroup$
@LutzL here's the LW scheme that I'm talking about : $h$ is the step we subdivide an interval $(a,b)$ into many $(t_i,t_{i+1})$ with $t_0 = a,,, t_{N} = b$, the eqn is $y' =f(t,y)$, $f$ is a 'nice' function then here's the algorithm : $$t_{n+1} = t_{n} + h,, 0 leq n leq N-1, ,, L_n = hf(t_n,y_n), ,, y_{n+1} = y_{n} +hf(t_n+frac{h}{2},y_n+frac{L_n}{2})$$ $y_{n+1}$ approximates $y(t_{n+1})$
$endgroup$
– rapidracim
Jan 6 at 18:24
1
$begingroup$
I know Lax-Wendroff as a method to approximate the Burgers equation and similar, Crank-Nicolson is usually introduced to solve the heat equation and similar. If you formulate it as method of lines, it reduces to a system of ODE, but that is not the primary problem statement.
$endgroup$
– LutzL
Jan 6 at 18:26
$begingroup$
That is the explicit midpoint method, or improved Euler method. It may be used in one of the variations of the LW scheme in the configuration of the time step. You then have probably the usual misconception to give the Heun, explicit trapezoidal, modified Euler method another name (because 3 names is not enough?) because the time step of CR is configured like the implicit trapezoidal method?
$endgroup$
– LutzL
Jan 6 at 18:29
|
show 2 more comments
$begingroup$
The second and third are methods for PDE. Is that the context you want to have considered in the references?
$endgroup$
– LutzL
Jan 6 at 18:13
$begingroup$
@LutzL Actually I only know them as methods for approximating solutions of ODE's
$endgroup$
– rapidracim
Jan 6 at 18:16
$begingroup$
@LutzL here's the LW scheme that I'm talking about : $h$ is the step we subdivide an interval $(a,b)$ into many $(t_i,t_{i+1})$ with $t_0 = a,,, t_{N} = b$, the eqn is $y' =f(t,y)$, $f$ is a 'nice' function then here's the algorithm : $$t_{n+1} = t_{n} + h,, 0 leq n leq N-1, ,, L_n = hf(t_n,y_n), ,, y_{n+1} = y_{n} +hf(t_n+frac{h}{2},y_n+frac{L_n}{2})$$ $y_{n+1}$ approximates $y(t_{n+1})$
$endgroup$
– rapidracim
Jan 6 at 18:24
1
$begingroup$
I know Lax-Wendroff as a method to approximate the Burgers equation and similar, Crank-Nicolson is usually introduced to solve the heat equation and similar. If you formulate it as method of lines, it reduces to a system of ODE, but that is not the primary problem statement.
$endgroup$
– LutzL
Jan 6 at 18:26
$begingroup$
That is the explicit midpoint method, or improved Euler method. It may be used in one of the variations of the LW scheme in the configuration of the time step. You then have probably the usual misconception to give the Heun, explicit trapezoidal, modified Euler method another name (because 3 names is not enough?) because the time step of CR is configured like the implicit trapezoidal method?
$endgroup$
– LutzL
Jan 6 at 18:29
$begingroup$
The second and third are methods for PDE. Is that the context you want to have considered in the references?
$endgroup$
– LutzL
Jan 6 at 18:13
$begingroup$
The second and third are methods for PDE. Is that the context you want to have considered in the references?
$endgroup$
– LutzL
Jan 6 at 18:13
$begingroup$
@LutzL Actually I only know them as methods for approximating solutions of ODE's
$endgroup$
– rapidracim
Jan 6 at 18:16
$begingroup$
@LutzL Actually I only know them as methods for approximating solutions of ODE's
$endgroup$
– rapidracim
Jan 6 at 18:16
$begingroup$
@LutzL here's the LW scheme that I'm talking about : $h$ is the step we subdivide an interval $(a,b)$ into many $(t_i,t_{i+1})$ with $t_0 = a,,, t_{N} = b$, the eqn is $y' =f(t,y)$, $f$ is a 'nice' function then here's the algorithm : $$t_{n+1} = t_{n} + h,, 0 leq n leq N-1, ,, L_n = hf(t_n,y_n), ,, y_{n+1} = y_{n} +hf(t_n+frac{h}{2},y_n+frac{L_n}{2})$$ $y_{n+1}$ approximates $y(t_{n+1})$
$endgroup$
– rapidracim
Jan 6 at 18:24
$begingroup$
@LutzL here's the LW scheme that I'm talking about : $h$ is the step we subdivide an interval $(a,b)$ into many $(t_i,t_{i+1})$ with $t_0 = a,,, t_{N} = b$, the eqn is $y' =f(t,y)$, $f$ is a 'nice' function then here's the algorithm : $$t_{n+1} = t_{n} + h,, 0 leq n leq N-1, ,, L_n = hf(t_n,y_n), ,, y_{n+1} = y_{n} +hf(t_n+frac{h}{2},y_n+frac{L_n}{2})$$ $y_{n+1}$ approximates $y(t_{n+1})$
$endgroup$
– rapidracim
Jan 6 at 18:24
1
1
$begingroup$
I know Lax-Wendroff as a method to approximate the Burgers equation and similar, Crank-Nicolson is usually introduced to solve the heat equation and similar. If you formulate it as method of lines, it reduces to a system of ODE, but that is not the primary problem statement.
$endgroup$
– LutzL
Jan 6 at 18:26
$begingroup$
I know Lax-Wendroff as a method to approximate the Burgers equation and similar, Crank-Nicolson is usually introduced to solve the heat equation and similar. If you formulate it as method of lines, it reduces to a system of ODE, but that is not the primary problem statement.
$endgroup$
– LutzL
Jan 6 at 18:26
$begingroup$
That is the explicit midpoint method, or improved Euler method. It may be used in one of the variations of the LW scheme in the configuration of the time step. You then have probably the usual misconception to give the Heun, explicit trapezoidal, modified Euler method another name (because 3 names is not enough?) because the time step of CR is configured like the implicit trapezoidal method?
$endgroup$
– LutzL
Jan 6 at 18:29
$begingroup$
That is the explicit midpoint method, or improved Euler method. It may be used in one of the variations of the LW scheme in the configuration of the time step. You then have probably the usual misconception to give the Heun, explicit trapezoidal, modified Euler method another name (because 3 names is not enough?) because the time step of CR is configured like the implicit trapezoidal method?
$endgroup$
– LutzL
Jan 6 at 18:29
|
show 2 more comments
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$begingroup$
The second and third are methods for PDE. Is that the context you want to have considered in the references?
$endgroup$
– LutzL
Jan 6 at 18:13
$begingroup$
@LutzL Actually I only know them as methods for approximating solutions of ODE's
$endgroup$
– rapidracim
Jan 6 at 18:16
$begingroup$
@LutzL here's the LW scheme that I'm talking about : $h$ is the step we subdivide an interval $(a,b)$ into many $(t_i,t_{i+1})$ with $t_0 = a,,, t_{N} = b$, the eqn is $y' =f(t,y)$, $f$ is a 'nice' function then here's the algorithm : $$t_{n+1} = t_{n} + h,, 0 leq n leq N-1, ,, L_n = hf(t_n,y_n), ,, y_{n+1} = y_{n} +hf(t_n+frac{h}{2},y_n+frac{L_n}{2})$$ $y_{n+1}$ approximates $y(t_{n+1})$
$endgroup$
– rapidracim
Jan 6 at 18:24
1
$begingroup$
I know Lax-Wendroff as a method to approximate the Burgers equation and similar, Crank-Nicolson is usually introduced to solve the heat equation and similar. If you formulate it as method of lines, it reduces to a system of ODE, but that is not the primary problem statement.
$endgroup$
– LutzL
Jan 6 at 18:26
$begingroup$
That is the explicit midpoint method, or improved Euler method. It may be used in one of the variations of the LW scheme in the configuration of the time step. You then have probably the usual misconception to give the Heun, explicit trapezoidal, modified Euler method another name (because 3 names is not enough?) because the time step of CR is configured like the implicit trapezoidal method?
$endgroup$
– LutzL
Jan 6 at 18:29