High Order Finite Difference Schemes for the Heat Equation Whose Convergence Rates are Higher Than Their Truncation Errors

Ditkowski, Adi

doi:10.1007/978-3-319-19800-2_13

Cited by 6 publications

(16 citation statements)

References 9 publications

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“…In [3], the scheme is formulated as V n+1 = QV n (2) where the operator Q is represented by the following Q = A + hBf and A, B ∈ R s×s . There are 4 sufficient conditions imposed on the matrices A and B in order to be error inhibiting:…”

Section: Error Inhibiting Block One-step Methodsmentioning

confidence: 99%

“…Among the large family of general linear methods the diagonally implicit multistage integration methods (DIMSIMs) in [1] are the special cases, which exhibit considerable potential for efficient implementation, providing the global error of the same order as the local truncation error. In [2], it was demonstrated that finite difference methods for PDEs can be constructed such that their convergence rates, or the order of their global errors, are higher than the order of the truncation errors. Following this idea, Ditkowski and Gottlieb devised the error inhibiting strategy in [3] by inhibiting the lowest order term in the truncation error from accumulating over time and thus showed that the global error of the scheme is one order higher than the local truncation error.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Adaptive Error Inhibiting Block One-Step Method for Ordinary Differential Equations

Jung

2020

Lecture Notes in Computational Science and Engineering

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An explicit error inhibiting block one-step method was developed by Ditkowski and Gottlieb in 2017. It is motivated by a class of one-step methods described in Shampine and Watts (Math Comp 23:731–740, 1969), which takes initial s step values to generate the next s step values at each step. The error inhibiting method shares the form of the diagonally implicit multistage integration method of type 3 introduced in Butcher (Appl Numer Math 11:347–363, 1993). With the explicit error inhibiting scheme, the convergence of the global error is one order higher than that of the local truncation error, while in general the global error decays with the same order as the local truncation error. In this work, we improve the explicit error inhibiting block one-step method in order to further enhance the convergence and accuracy of the method. The main idea is to adopt the radial basis functions as a reconstruction basis replacing the polynomial basis. The numerical results confirm that the proposed method is efficient in providing enhanced order of accuracy.

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Section: Error Inhibiting Block One-step Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Adaptive Error Inhibiting Block One-Step Method for Ordinary Differential Equations

Jung

2020

Lecture Notes in Computational Science and Engineering

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“…The structure of this scheme, and a similar scheme with first-order truncation error and third-order convergence rate (4th order with post-processing) was assumed in [7], [9]. Further details on the derivation of the scheme are provided in Appendix C.…”

Section: Block Finite Difference Schemesmentioning

confidence: 99%

“…In this section, our goal is to utilize the FE methodology for proving the stability of the BFD scheme. In [7] and [9]. he stability was proven using a Fourier-like analysis.…”

Section: Proof Of Stability 221 Motivationmentioning

confidence: 99%

Error Inhibiting Methods for Finite Elements

Ditkowski,

Blanc,

Shu

2020

Preprint

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Finite Difference methods (FD) are one of the oldest and simplest methods used for solving differential equations. Theoretical results have been obtained during the last six decades regarding the accuracy, stability and convergence of the FD method for partial differential equations (PDE).The local truncation error is defined by applying the difference operator to the exact solution u. In the classical FD method, the orders of the global error and the truncation error are the same.Block Finite Difference methods (BFD) are difference methods in which the domain is divided into blocks, or cells, containing two or more grid points with a different scheme used for each grid point, unlike the standard FD method. In this approach, the interaction between the different truncation errors and the dynamics of the scheme may prevent the error from growing, hence error reduction is obtained. The phenomenon in which the order of the global error is smaller than the one of the truncation error is called error inhibition, see e.g. [8].The Finite Element method (FE) consists in finding an approximation of the solution in a certain form, usually a linear combination of a set of chosen trial functions, i.e.q j ϕ j . Since, in most cases, the exact solution will not lie in the space spanned by those functions, the coefficients of the

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“…There are many methods, for example some meshless methods, that are difficult to classify and blur the distinction between finite-difference and finite-element methods. An interesting recent example can be found in Ditkowski [6], where he introduces finite-difference operators with two-and three-point blocks.…”

Section: Introductionmentioning

confidence: 99%

Opportunities for efficient high-order methods based on the summation-by-parts property (Invited)

Hicken

Fernández

Zingg

2015

22nd AIAA Computational Fluid Dynamics Conference

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Summation-by-parts (SBP) operators are traditionally viewed as high-order finite-difference operators, but they can also be interpreted as finite-element operators with an implicit basis. Such an element-based perspective leads to several opportunities that we describe. The first is provided by generalized one-dimensional SBP operators, which maintain the desirable properties of classical SBP operators while permitting flexible nodal distributions. The second opportunity is to extend the SBP definition to multiple dimensions, and a recently proposed definition for multidimensional SBP operators paves the way for time-stable, high-order finite-difference operators on unstructured grids. The final opportunity that we discuss is an analogy with the continuous Galerkin finite-element method, which leads to a systematic means of assembling SBP operators on a global domain from elemental operators. To illustrate these ideas, high-order SBP operators are constructed for the triangle and tetrahedron, and the former are assembled into a global SBP operator and applied to a linear convection problem on a triangular grid.

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High Order Finite Difference Schemes for the Heat Equation Whose Convergence Rates are Higher Than Their Truncation Errors

Cited by 6 publications

References 9 publications

An Adaptive Error Inhibiting Block One-Step Method for Ordinary Differential Equations

An Adaptive Error Inhibiting Block One-Step Method for Ordinary Differential Equations

Error Inhibiting Methods for Finite Elements

Opportunities for efficient high-order methods based on the summation-by-parts property (Invited)

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