High-order transverse schemes for the numerical solution of PDEs

Mazzia, Annamaria; Mazzia, Francesca

doi:10.1016/s0377-0427(97)00090-3

Cited by 13 publications

(10 citation statements)

References 22 publications

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“…The first problem we have considered is an artificial one due to Cash [20], which has been used as a test problem different times [21,22]. The problem is a form of the heat equation that describes heat flow in an insulated thin rod with zero temperature at the edges, and is given by…”

Section: A Linear Parabolic Pdementioning

confidence: 99%

“…as it was done in [22], where u ij is the numerical solution at time t j and space x i , and u(x i , t j ) is the exact solution. The results with the method of third order in the previous sections are presented in Table I where we have included between brackets the corresponding results for the t and x values obtained with the longitudinal scheme of fourth order in Mazzia and Mazzia [22].…”

Section: A Linear Parabolic Pdementioning

confidence: 99%

“…The results with the method of third order in the previous sections are presented in Table I where we have included between brackets the corresponding results for the t and x values obtained with the longitudinal scheme of fourth order in Mazzia and Mazzia [22].…”

Section: A Linear Parabolic Pdementioning

confidence: 99%

See 2 more Smart Citations

An almost L‐stable BDF‐type method for the numerical solution of stiff ODEs arising from the method of lines

Ramos

Vigo‐Aguiar

2007

Numerical Methods Partial

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A new BDF-type scheme is proposed for the numerical integration of the system of ordinary differential equations that arises in the Method of Lines solution of time-dependent partial differential equations. This system is usually stiff, so it is desirable for the numerical method to solve it to have good properties concerning stability. The method proposed in this article is almost L-stable and of algebraic order three. Numerical experiments illustrate the performance of the new method on different stiff systems of ODEs after discretizing in the space variable some PDE problems.

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Section: A Linear Parabolic Pdementioning

confidence: 99%

Section: A Linear Parabolic Pdementioning

confidence: 99%

See 1 more Smart Citation

An almost L‐stable BDF‐type method for the numerical solution of stiff ODEs arising from the method of lines

Ramos

Vigo‐Aguiar

2007

Numerical Methods Partial

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“…Both of such approaches have been utilized in numerical analysis. The former is used, for example, in [3,5,13,14]. The latter has been used in the past, for example by Dahlquist, in studying the stability problem for numerical methods [8].…”

Section: A Characterization Of Positive Definite Band Symmetric Toeplmentioning

confidence: 99%

“…Moreover, the relations between the well-conditioning of the semi-discrete boundary value problem, arising by applying the transversal method of lines, and the spectrum have been studied in [14].…”

Section: Introductionmentioning

confidence: 99%

On the properties of matrices defining some classes of BVMs

Aceto

Trigiante

2003

Numerische Mathematik

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It is known that the matrices defining the discrete problem generated by a k-step Boundary Value Method (BVM) have a quasi-Toeplitz band structure. In particular, when the boundary conditions are skipped, they become Toeplitz matrices. In this paper, by introducing a characterization of positive definiteness for such matrices, we shall prove that the Toeplitz matrices which arise when using the methods in the classes of BVMs known as Generalized BDF and Top Order Methods have such property

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