Combined compact difference scheme for linear second-order partial differential equations with mixed derivative

Lee, Spike T.; Li, Jun; Sun, Hai-Wei

doi:10.1016/j.cam.2014.01.004

Cited by 20 publications

(32 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Equation (1) is widely used in the fields of porous media flow [2,3] and when coordinate transformations are applied to a convection-diffusion equation on non-rectangular domains [4], which also generate Eq. (1). So it is both theoretically and practically important to investigate their accurate, stable and efficient numerical methods.…”

Section: Introductionmentioning

confidence: 99%

“…So it is both theoretically and practically important to investigate their accurate, stable and efficient numerical methods. In the past decades, high-order compact (HOC) difference methods have attracted increasing interests for solving such equations [1,[5][6][7][8]. The main feature of HOC methods is that they have higher-order accuracy (usually fourth-order, even higher) and higher spectral resolution with relatively few grid points (compact grid stencil) than the conventional difference methods.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A higher-order blended compact difference (BCD) method for solving the general 2D linear second-order partial differential equation

2019

Adv Differ Equ

View full text Add to dashboard Cite

A higher-order blended compact difference (BCD) scheme is proposed to solve the general two-dimensional (2D) linear second-order partial differential equation. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. Sixth-order accuracy approximations for the first-and second-order derivatives are employed, and the original equation is also discretized based on a 9-point stencil, which is different from the work of Lee et al. (J. Comput. Appl. Math. 264:23-37, 2014). A truncation error analysis is performed to show that the scheme is of sixth-order accuracy for the interior grid points. Simultaneously, sixth-order accuracy schemes are proposed to compute the grid points on the boundaries for the first-and second-order derivatives. Numerical experiments are conducted to demonstrate the accuracy and efficiency of the present method.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A higher-order blended compact difference (BCD) method for solving the general 2D linear second-order partial differential equation

2019

Adv Differ Equ

View full text Add to dashboard Cite

show abstract

“…Circulant matrices play an important role in solving many different differential equations, such as ordinary, partial, matrix, linear second-order partial, bi-Hamiltonian partial, parameterized delay, fractional order, and singular perturbation delay. Lee et al investigated a high-order compact (HOC) scheme for the general two-dimensional (2D) linear partial differential equation in [1] with a mixed derivative. Meanwhile, in order to establish the CCD2 scheme, they rewrote equation (1.1) into (2.1) in [1].…”

Section: Introductionmentioning

confidence: 99%

“…Lee et al investigated a high-order compact (HOC) scheme for the general two-dimensional (2D) linear partial differential equation in [1] with a mixed derivative. Meanwhile, in order to establish the CCD2 scheme, they rewrote equation (1.1) into (2.1) in [1]. To write the CCD2 system in a concise style, they employed circulant matrix to obtain the corresponding whole CCD2 linear system (2.10), whose entries are circulant block.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Minimal Polynomials and the Inverses of Multilevel Scaled Factor Circulant Matrices

Jiang

2014

Abstract and Applied Analysis

View full text Add to dashboard Cite

Circulant matrices have important applications in solving various differential equations. The level-kscaled factor circulant matrix over any field is introduced. Algorithms for finding the minimal polynomial of this kind of matrices over any field are presented by means of the algorithm for the Gröbner basis of the ideal in the polynomial ring. And two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for computing the inverse of partitioned matrix with level-kscaled factor circulant matrix blocks over any field is given by using the Schur complement, which can be realized by CoCoA 4.0, an algebraic system, over the field of rational numbers or the field of residue classes of modulo prime number.

show abstract

Two‐sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems

Pultarová

Ladecký

2021

Numerical Linear Algebra App

View full text Add to dashboard Cite

Numerical methods for elliptic partial differential equations usually lead to systems of linear equations with sparse, symmetric, and positive definite matrices. In many methods, these matrices can be obtained as sums of local symmetric positive semidefinite matrices. In this article, we use this assumption and introduce a method that provides guaranteed lower and upper bounds to all individual eigenvalues of the preconditioned matrices. We apply the method for preconditioners arising from the same discretization problem but with simplified coefficients. The method uses solely the data over the solution domain and local connections between the degrees of freedom defined by the discretization.

show abstract

Combined compact difference scheme for linear second-order partial differential equations with mixed derivative

Cited by 20 publications

References 19 publications

A higher-order blended compact difference (BCD) method for solving the general 2D linear second-order partial differential equation

A higher-order blended compact difference (BCD) method for solving the general 2D linear second-order partial differential equation

On the Minimal Polynomials and the Inverses of Multilevel Scaled Factor Circulant Matrices

Two‐sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems

Contact Info

Product

Resources

About