Transformed implicit-explicit second derivative diagonally implicit multistage integration methods with strong stability preserving explicit part

Moradi, Afsaneh; Sharifi, Mohammad Bagher; Abdi, Ali

doi:10.1016/j.apnum.2020.04.007

Cited by 12 publications

(8 citation statements)

References 37 publications

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“…By using the results from [13, 27, 41], sufficient conditions for an SGLM to preserve the strong stability properties of spatial discretizations, when coupled with forward Euler and a second derivative formulation, were determined by Moradi et al [36]. To describe a numerical search for SSP SGLMs, the authors [36] considered the constrained minimization problem with an objective function subject to the following nonlinear inequality constraints: where is some constant, can take any positive value and These conditions are equivalent to For an SGLM with the coefficient matrices and the abscissa vector c , the SSP coefficient was given by Moradi et al [36] as Considering these conditions leads to a wide variety of SSP SGLMs [34–36, 38], where the main drawback of these conditions is that the types of spatial discretizations that can be used are limited. Due to this reason, in this paper, we will consider a different approach to the SSP concept, so that there will be more flexibility in the choice of spatial discretizations.…”

Section: A Short Review On the Ssp Sglmsmentioning

confidence: 99%

“…Using this type of condition, Christlieb et al [13] derived SSP two-derivative RK methods up to order six, which preserve the strong stability properties of the forward Euler condition (1.3) when coupled with the second derivative condition (1.4). Moradi et al [36], using the general order conditions for second derivative general linear methods (SGLMs) derived in [37], extended this SSP approach to characterize and design SSP SGLMs as a class of multistage second derivative time discretization and investigated more in [34, 35, 38]. The main flaw of this approach is that the types of spatial discretizations that can be used are limited.…”

Section: Introductionmentioning

confidence: 99%

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High Order Explicit Second Derivative Methods With Strong Stability Properties Based on Taylor Series Conditions

2022

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When faced with the task of solving hyperbolic partial differential equations (PDEs), high order, strong stability-preserving (SSP) time integration methods are often needed to ensure preservation of the nonlinear strong stability properties of spatial discretizations. Among such methods, SSP second derivative time-stepping schemes have been recently introduced and used for evolving hyperbolic PDEs. In previous works, coupling of forward Euler and a second derivative formulation led to sufficient conditions for a second derivative general linear method (SGLM), which preserve the strong stability properties of spatial discretizations. However, for such methods, the types of spatial discretizations that can be used are limited. In this paper, we use a formulation based on forward Euler and Taylor series conditions to extend the SSP SGLM framework. We investigate the construction of SSP second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of SGLMs with order $p=r=s$ and stage order $q=p,p-1$ up to order eight, where r is the number of external stages and s is the number of internal stages of the method. Proposed methods are examined on some one-dimensional linear and nonlinear systems to verify their theoretical order, and show potential of these schemes in preserving some nonlinear stability properties such as positivity and total variation.

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Section: A Short Review On the Ssp Sglmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High Order Explicit Second Derivative Methods With Strong Stability Properties Based on Taylor Series Conditions

2022

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“…u(x) = r(k(2x − 1) sinh(kx) + 2 cosh(kx) tanh( k 2 )) 2k 3 is the exact solution of example 5.1, and the numerical results are given in Table 1. u (4) − 2u = −1 − (8π 4 − 1) cos(2πx), x ∈ (0, 1) u(0) = u(1) = u (0) = u (1) = 0. Table 2 and figure 2 lists the errors comparison between the multilevel augmentation method [30] and our method for this BVPs…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…It is difficult to find the analytic solutions of higher-order BVPs because of the complexity of the systems, many numerical algorithms for high-order BVPs have been proposed in recent years. The multistage integration method is an important method to solve the numerical solution of higher-order models by reducing the order gradually [2][3][4][5]. Cao [6] solved a class of higher-order fractional ordinary differential equations by the quadratic interpolation function method.…”

Section: Introductionmentioning

confidence: 99%

Multiscale Method for Solving High-order BVPs

Zhang¹,

Mei²,

Yingzhen³

2020

Preprint

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This paper presents a numerical algorithm for solving high-order BVPs. We introduce the construction method of multiscale orthonormal basis in Wm[0; 1] by multiscale orthonormal basis in W1[0; 1]. We define approximate solution, and obtain the approximate solution of high-order BVPs by using the approximate theory. Moreover, the convergence and stability of the algorithm are improved. At last, several numerical experiments show the feasibility of the proposed method.

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“…Por otra parte, los métodos implícitos para solucionar ecuaciones diferenciales ordinarias tienden a tener regiones de estabilidad que contienen a gran parte del plano complejo, lo cual permite su implementación en sistemas diferenciales tipo Stiff [10,13]. A pesar de su naturaleza, su ejecución es bastante costosa computacionalmente, ya que implica la solución en simultaneo de un número elevado de ecuaciones [10,14].…”

Section: Introductionunclassified

Una aproximación numérica a los General Linear Methods para la solución de problemas de la cinética química

Niño

Polo

Candezano

2021

Investigación e Innovación en Ingenierías

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Objetivo: Realizar un estudio comparativo entre las soluciones numéricas de sistemas de ecuaciones diferenciales ordinarias tipo Stiff obtenidas con el método General Linear Method (GLM) y dos solvers del software matemático MATLAB como ode45 y ode15s. Metodología: Para el estudio comparativo se escogieron dos problemas de la química HIRES y de OREGO, los cuales modelan reacciones de la cinética química. Inicialmente cada problema fue solucionado utilizando un tipo de GLM. De igual forma, fueron calculadas las soluciones numéricas con los solvers ode45 y ode15s del software MATLAB. Finalmente, se establecieron comparaciones teniendo en cuenta el tiempo de ejecución de cada método/solver para resolver el problema, el número de iteraciones totales y el error relativo. Resultados: A partir de los resultados numéricos del problema de HIRES se observó que el método DIMSIM tuvo el mejor rendimiento comparando los tiempos de CPU, pero el solver ode45 de MATLAB tuvo la mejor aproximación, seguido por ode15s y los obtenidos por el método DIMSIM, respectivamente. Con el problema OREGO, el mejor tiempo computacional fue alcanzado por el solver ode15s, sin embargo, los resultados obtenidos con el método DIMSIM tuvieron la mejor aproximación. Conclusiones: Los resultados numéricos indicaron que el método DIMSIM y el solver ode15s obtuvieron un mejor desempeño en el tiempo de ejecución y el número de iteraciones totales comparados con el solver ode45. Para estudios futuros se pretende comparar los resultados numéricos con aquellos obtenidos con estrategias de paralelización del código computacional.

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Transformed implicit-explicit second derivative diagonally implicit multistage integration methods with strong stability preserving explicit part

Cited by 12 publications

References 37 publications

High Order Explicit Second Derivative Methods With Strong Stability Properties Based on Taylor Series Conditions

High Order Explicit Second Derivative Methods With Strong Stability Properties Based on Taylor Series Conditions

Multiscale Method for Solving High-order BVPs

Una aproximación numérica a los General Linear Methods para la solución de problemas de la cinética química

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