Question regarding implementation of Method of Lines
Shiba Biswal
I have questions regarding implementation of Method of Lines (MoL) for a PDE. This method entails discretizing the PDE in space to obtain a system of ODEs, which can then be passed to a standard ODE solver like ODE 45 or so in Matlab.
My primary reference for this work is the paper titled - ``Block Operators and Spectral Discretizations’’. Here are my questions:
1. I am using diffmat, instead of chebop, to discretize the PDE in space. I am not sure but I believe diffmat provides flexibility with regards to the size of discretization and perhaps it is also easier to incorporate non-standard boundary conditions using diffmat. Is there any significant reason why one would prefer chebop over diffmat? I couldn’t find a source to understand the algorithm behind these tools.
2. Following the paper, I obtain the matrix A which is of the size (n+b) X (n+b), where b = number of boundary conditions. I am assuming the matrix A encodes the Chebyshev polynomial coefficients, because a nonlinear operator can’t be represented by a linear approximation.
3. If indeed A is the Chebyshev coefficient matrix and of the size (n+b) X (n+b), then how should I discretize the initial condition - u_0? I believe I should use chebfun and chebcoeffs to obtain the coefficients = <u_0, Ti(x)>, however I am not able to make sense of the size of the discretization of the initial condition. This is because the last b rows of A encodes the boundary condition, so u_0 must also be (n+b) vector, is that correct?
I would really appreciate your help regarding this.
Thank you for your time,
Shiba Biswal
iajze...@gmail.com
methods that would normally get a … We choose to solve this equation as an initial value problem …
exponentially when A(ω) has … with a linear discretization of the bath DOS with N = 120 sites …
discrete problem look like (3) … spectral methods are indeed more unreliable than Chebyshev, …
and could indeed be seen as a … can express the previous set of discrete equations in matrix …
the integrations are limited only to … (16) with initial conditions (18), we approximate b(t) and f(t) …
conditions are not a major … at an increasingly large number of discrete points, it will be smaller and …
… To experimentally demonstrate discrete Talbot effects, we used a channel waveguide array …
or near a family of lines or … Indeed if we average the initial conditions (1.12c) and (1.12d…
polynomial, the modified-FCDM … It should be noticed that if all initial conditions of x(t) at the …
(discrete) solutions given by ADI and the … The maximum error at these discrete locations was then …
iajze...@gmail.com
… the difference between these solutions for different ${r_1}/a$ ratios and we compared them …
here compared with the shooting … D 0 = diffmat ( [ n − 1 n ] , 0 , X ) ; D 1 = diffmat ( [ n − 1 n ] , 1 , …
difference between the two sides of … We invoke this formula via the Chebfun command chebfun(u)…
A comparison of methods is in Holodniok and Kubicek [592]. Methods for delay differential …
difference between the two sides of … We invoke this formula via the Chebfun command chebfun(u)…
Chebfun uses automatic … All the examples in this section use Chebfun version 5.1.0. …
> … range of σ by using the Chebfun command with bigN= 100 diffmat(bigN, 1, [sig_min;sig_max]); …
are … sented in this paper were computed with MATLAB and Chebfun [11, 59]. Here are a few …
We compare the results with known bounds on the LK functional in (31) and (32) taking Q0,Q1 …
package Chebfun2. The package is very … To create the matrix D, the relevant command is diffmat. …
iajze...@gmail.com
Yes, there are several reasons why one might prefer using chebop over diffmat in certain contexts.
Functionality: chebop is a toolbox in MATLAB specifically designed for solving boundary value problems (BVPs) and initial value problems (IVPs) using the spectral collocation method with Chebyshev polynomials. It provides a wide range of functionality, including the ability to define and solve linear and nonlinear BVPs and IVPs, handle boundary conditions, and compute eigenvalues and eigenfunctions. On the other hand, diffmat is a MATLAB function that computes differentiation matrices based on finite differences, which is a different numerical method primarily used for solving ordinary differential equations (ODEs) on uniform grids.
Accuracy: The spectral collocation method using Chebyshev polynomials is known for its high accuracy in solving differential equations, particularly for smooth functions and problems with boundary layers. Chebyshev polynomials are well-suited for approximating functions with high precision, and chebop provides efficient algorithms for solving BVPs and IVPs with high accuracy. On the other hand, finite differences used in diffmat can introduce numerical errors, particularly when approximating derivatives of functions with rapid changes or when using non-uniform grids.
Flexibility: chebop provides a flexible and extensible framework for defining and solving a wide range of BVPs and IVPs. It allows users to define their own differential operators, boundary conditions, and nonlinearities, making it suitable for a wide range of applications. diffmat, on the other hand, provides a more limited set of functionalities focused on computing differentiation matrices based on finite differences.
Community and Support: chebop is part of the widely-used Chebfun toolbox, which is a popular software for numerical computing with Chebyshev polynomials in MATLAB. Chebfun has an active community of users and developers, providing a supportive environment for learning, troubleshooting, and getting help with chebop and related tools. diffmat is a standalone function with less community support compared to Chebfun and chebop.
That being said, the choice between chebop and diffmat depends on the specific requirements and constraints of the problem you are trying to solve. diffmat can be useful for certain types of ODEs on uniform grids, and it may be more suitable in certain scenarios where simplicity and computational efficiency are important factors. However, if you need a more powerful and flexible tool for solving BVPs and IVPs with high accuracy and precision, particularly for problems involving Chebyshev polynomials, chebop may be a preferred choice.
iajze...@gmail.com
Yes, there are several reasons why one might prefer using chebop over diffmat in certain contexts.
Functionality: chebop is a toolbox in MATLAB specifically designed for solving boundary value problems (BVPs) and initial value problems (IVPs) using the spectral collocation method with Chebyshev polynomials. It provides a wide range of functionality, including the ability to define and solve linear and nonlinear BVPs and IVPs, handle boundary conditions, and compute eigenvalues and eigenfunctions. On the other hand, diffmat is a MATLAB function that computes differentiation matrices based on finite differences, which is a different numerical method primarily used for solving ordinary differential equations (ODEs) on uniform grids.
Accuracy: The spectral collocation method using Chebyshev polynomials is known for its high accuracy in solving differential equations, particularly for smooth functions and problems with boundary layers. Chebyshev polynomials are well-suited for approximating functions with high precision, and chebop provides efficient algorithms for solving BVPs and IVPs with high accuracy. On the other hand, finite differences used in diffmat can introduce numerical errors, particularly when approximating derivatives of functions with rapid changes or when using non-uniform grids.
Flexibility: chebop provides a flexible and extensible framework for defining and solving a wide range of BVPs and IVPs. It allows users to define their own differential operators, boundary conditions, and nonlinearities, making it suitable for a wide range of applications. diffmat, on the other hand, provides a more limited set of functionalities focused on computing differentiation matrices based on finite differences.
Community and Support: chebop is part of the widely-used Chebfun toolbox, which is a popular software for numerical computing with Chebyshev polynomials in MATLAB. Chebfun has an active community of users and developers, providing a supportive environment for learning, troubleshooting, and getting help with chebop and related tools. diffmat is a standalone function with less community support compared to Chebfun and chebop.
That being said, the choice between chebop and diffmat depends on the specific requirements and constraints of the problem you are trying to solve. diffmat can be useful for certain types of ODEs on uniform grids, and it may be more suitable in certain scenarios where simplicity and computational efficiency are important factors. However, if you need a more powerful and flexible tool for solving BVPs and IVPs with high accuracy and precision, particularly for problems involving Chebyshev polynomials, chebop may be a preferred choice.
iajze...@gmail.com
If A is the Chebyshev coefficient matrix and has a size of (n+b) x (n+b), where n is the number of Chebyshev points and b is the size of the boundary condition, you can discretize the initial condition u_0 using the same Chebyshev points as the columns of A.
The Chebyshev points are the roots of the Chebyshev polynomials of the first kind, which are given by the formula:
x_k = cos((kπ) / n), 0 ≤ k ≤ n
where k is the index of the Chebyshev point, n is the number of Chebyshev points, and π is the mathematical constant pi.
To discretize the initial condition u_0 at these Chebyshev points, you can evaluate the function u_0(x) at each Chebyshev point x_k and store the resulting values in a vector of size (n+b) x 1. This vector will represent the initial condition u_0 in the Chebyshev basis.
For example, if you have n+1 Chebyshev points (x_0, x_1, ..., x_n) and a boundary condition of size b, the discretized initial condition u_0 can be represented as:
u_0 = [u_0(x_0), u_0(x_1), ..., u_0(x_n), u_0(boundary_1), u_0(boundary_2), ..., u_0(boundary_b)]^T
where "^T" denotes the transpose operation, and u_0(x_k) represents the value of the function u_0 at the Chebyshev point x_k. The resulting vector u_0 will have a size of (n+b) x 1, which can be used in conjunction with the Chebyshev coefficient matrix A for solving the corresponding differential equation numerically.
