Optimal orders of convergence for Runge–Kutta methods and linear, initial boundary value problems

Alonso‐Mallo, Isaías; Palencia, C.

doi:10.1016/s0168-9274(02)00110-1

Cited by 21 publications

(54 citation statements)

References 15 publications

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“…The result is that the temporal mismatch becomes a spatial numerical boundary layer. Suggestions for mitigating this type of order reduction have been proposed for explicit Runge-Kutta schemes [52,15,4,3]. Since SDC methods also employ lower-order solutions, they are also susceptible to this type of order reduction.…”

Section: : End Formentioning

confidence: 99%

“…The cause of this type of order reduction is a mismatch in the error between the domain boundary and its interior in the stage values computed during the Runge-Kutta procedure. Various approaches for mitigating this type of order reduction have been proposed for explicit Runge-Kutta schemes [52,15,4,3], but no general remedy for non-linear PDEs has been developed. SISDC methods are also susceptible to order reduction in the presence of time-dependent boundary conditions because they also use lower-order stage values in the solution process [46].…”

Section: Introductionmentioning

confidence: 99%

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Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Minion

Saye

2018

Journal of Computational Physics

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This paper compares and contrasts higher-order, semi-implicit temporal integration strategies for the incompressible Navier-Stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations. Particular focus is placed on the imposition of boundary conditions in the semi-implicit formulation, the accurate treatment of the pressure term, and the smoothness of the numerical solution for different formulations. The main result presented here is the formulation and numerical validation of a single-step, semi-implicit method called spectral deferred pressure corrections, which can in theory obtain arbitrary formal order of accuracy in time. Numerical results demonstrate up to eighth-order accuracy, scenarios where optimal temporal accuracy is attained, and scenarios where order reduction is observed due to time-dependent boundary conditions.

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Section: : End Formentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Minion

Saye

2018

Journal of Computational Physics

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“…In our context, and taking advantage of our limitation to methods with p ≥ q + 1, the key theorem of [3] is Theorem 2. This result is stated for problems with homogeneous boundary conditions, but Section 4 of [3] explains how to handle the non-homogeneous boundary conditions and why the result still holds. The following translation table 1 4 − ε} 1 0 can be used to navigate [3,Theorem 2] and relate it to our particular problem.…”

Section: Error Estimatesmentioning

confidence: 99%

“…This result is stated for problems with homogeneous boundary conditions, but Section 4 of [3] explains how to handle the non-homogeneous boundary conditions and why the result still holds. The following translation table 1 4 − ε} 1 0 can be used to navigate [3,Theorem 2] and relate it to our particular problem. Note that when p ≥ q + 2, it is important to have D ⊂ V ⊂ [H, D(A)] 1/4−ε with bounded embeddings.…”

Section: Error Estimatesmentioning

confidence: 99%

Time-domain boundary integral equation modeling of heat transmission problems

et al. 2019

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This paper investigates the numerical modeling of a time-dependent heat transmission problem by the convolution quadrature boundary element method. It introduces the latest theoretical development into the error analysis of the numerical scheme. Semigroup theory is applied to obtain stability in spatial semidiscrete scheme. Functional calculus is employed to yield convergence in the fully discrete scheme. In comparison to the traditional Laplace domain approach, we show our approach gives better estimates.

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“…Such reductions have been deeply studied (see for example [4], [6], [22], [26]) and recently, Alonso-Mallo and Cano ( [2], [3]) have developed and analyzed a technique which can be used in Runge-Kutta or Rosenbrock methods to avoid such order reduction. This technique also consists of modifying cleverly the boundary conditions naturally associated with the internal stages of these methods.…”

Section: Introductionmentioning

confidence: 99%

Spectral-fractional step Runge–Kutta discretizations for initial boundary value problems with time dependent boundary conditions

Alonso‐Mallo

Cano

Jorge³

2004

Math. Comp.

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Abstract. In this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting, alternating direction or, more generally, fractional step methods. Such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive Runge-Kutta method, which is called the fractional step Runge-Kutta method, and a standard space discretization which can be of type finite differences, finite elements or spectral methods among others. Spectral methods have been chosen here to complete the analysis of convergence of a totally discrete scheme of this type of improved fractionary steps. The numerical experiences performed also show the increase of accuracy that this technique provides.

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Optimal orders of convergence for Runge–Kutta methods and linear, initial boundary value problems

Cited by 21 publications

References 15 publications

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Time-domain boundary integral equation modeling of heat transmission problems

Spectral-fractional step Runge–Kutta discretizations for initial boundary value problems with time dependent boundary conditions

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