Non-Ocillatory Limited-Time Integration for Conservation Laws and Convection-Diusion Equations Jingcheng Lu and James D.Baeder

2025-05-02 0 0 943.99KB 17 页 10玖币

侵权投诉

Non-Ocillatory Limited-Time Integration for Conservation Laws and

Convection-Diﬀusion Equations

Jingcheng Lu and James D.Baeder

Abstract

In this study we consider unconditionally non-oscillatory, high order implicit time marching based on time-

limiters. The ﬁrst aspect of our work is to propose the high resolution Limited-DIRK3 (L-DIRK3) scheme for

conservation laws and convection-diﬀusion equations in the method-of-lines framework. The scheme can be used

in conjunction with an arbitrary high order spatial discretization scheme such as 5th order WENO scheme. It

can be shown that the strongly S-stable DIRK3 scheme is not SSP and may introduce strong oscillations under

large time step. To overcome the oscillatory nature of DIRK3, the key idea of L-DIRK3 scheme is to apply local

time-limiters (K.Duraisamy, J.D.Baeder, J-G Liu), with which the order of accuracy in time is locally dropped to

ﬁrst order in the regions where the evolution of solution is not smooth. In this way, the monotonicity condition

is locally satisﬁed, while a high order of accuracy is still maintained in most of the solution domain. For

convenience of applications to systems of equations, we propose a new and simple construction of time-limiters

which allows ﬂexible choice of reference quantity with minimal computation cost. Another key aspect of our

work is to extend the application of time-limiter schemes to multidimensional problems and convection-diﬀusion

equations. Numerical experiments for scalar/systems of equations in one- and two-dimensions conﬁrm the high

resolution and the improved stability of L-DIRK3 under large time steps. Moreover, the results indicate the

potential of time-limiter schemes to serve as a generic and convenient methodology to improve the stability of

arbitrary DIRK methods.

1 Introduction

We consider high resolution, non-oscillatory implicit time integration schemes for hyperbolic conservation laws

and convection-diﬀusion problems. The most common approach of ﬁnding numerical solutions to time-dependent

partial diﬀerential equations is the method-of-lines. In this framework, we perform proper spatial discretization

over the suitable domain and then integrate the resulting system of ordinary diﬀerential equations (ODEs) using

standard time integration schemes. In many practical applications this may result in a stiﬀ ODE system, implicit

time integration may be preferred in order to enlarge the allowable time step.

Most of the existing studies on non-oscillatory numerical schemes focus on spatial discretizations. The early

stage framework was the class of monotone schemes, which, however, are only ﬁrst order accurate [8]. Based

on less restrictive stability conditions, such as Total Variation Diminishing (TVD) and essentially non-oscillatory,

many high order non-oscillatory spatial reconstructions have been proposed, such as the TVD schemes [6] and the

ENO/WENO schemes [16]. However, ensuring non-oscillatory solutions of these high order schemes may require a

severe restriction on the time step. Particularly, for high order implicit schemes this restriction can be much more

restrictive than their linear stability limits.

To enable the use of larger time step for high order implicit time marching, K.Duraisamy, J.D.Baeder and

J-G.Liu [4] proposed the use of time-limiters, with which the strong oscillations of trapezoidal scheme and DIRK2

scheme were eﬀectively reduced under large time steps. To better preserve the high resolution of high order spatial

reconstructions, in the present work we will extend the idea of limited-time integration to construct the new

Limited-DIRK3 (L-DIRK3). The key ingredient is to locally employ implicit Euler near discontinuities. A series

of numerical examples are presented to verify the improved stability and the high resolution of L-DIRK3.

2 Motivation of Time-Limiter Schemes

Consider initial value problem for conservation law

∂

∂t u(x, t) + ∂

∂x f(u(x, t)) = 0, u(x, 0) = u0(x), x ∈Ω, t ∈R+,(2.1)

arXiv:2210.14075v1 [math.NA] 25 Oct 2022

where uis a scalar/vector of conservative variables, fis a convective ﬂux. To numerically solve (2.1), we consider

the semidiscrete ﬁnite diﬀerence/ﬁnite volume schemes: at ﬁrst we perform proper spatial discretization, which

results in a set of ordinary diﬀerential equations

dt uj(t) +

fj+1

2−ˆ

fj−1

∆x= 0, j = 1,2,· · · , N. (2.2)

Here Nis the number of mesh points, uj(t) is the approximate solution to the point value u(xj, t) /the cell average

¯u(xj, t) := 1

∆xZxj+1

xj−1

u(x, t)dx,ˆ

fj+1

2is the numerical ﬂux at cell interface xj+1

2. Then we integrate (2.2) with

standard time integration schemes, e.g. Runge Kutta methods, linear multistep methods. While explicit time

integration is convenient to implement, in certain applications, e.g. steay-state computations and convection-

diﬀusion problems, one may prefer implicit time integration in order to apply a large time step and improve the

eﬃciency.

For nonlinear conservation laws, the solutions may develop discontinuities even if the initial data is smooth. To

prevent numerical oscillations near discontinuities, many high order non-oscillatory schemes have been proposed

in the last few decades and successfully applied in hyperbolic problems, including the UNO schemes [7], the MP

schemes [17], and the ENO/WENO type schemes [16], etc. However, these high order schemes are non-oscillatory

only under restrictive time steps. Gottlieb, Shu and Tadmor [5] have shown that when the order of accuracy in time

is higher than one, a time integration method, even if implicit, is at most conditionally strong stability preserving

(SSP). The time step restriction deteriorates the purpose of using of implicit methods in practical applications.

In order to use larger time step for high order implicit time integration without introducing severe oscillations,

K.Duraisamy, J. Baeder and J-G. Liu [4] proposed the time-limiter schemes. These schemes are obtained from

local convex combinations of a ﬁrst order unconditionally SSP method and a higher order but oscillatory method.

The idea is that in the regions of large solution gradients the scheme is switched to the ﬁrst order method, while

in the smooth regions we still apply the higher order time integration. Locally deﬁned time-limiters are introduced

to detect the occurrence of high gradients and determine the switch between diﬀerent methods. Such idea was

applied to construct the Limited Trapezoidal (L-Trap) and the Limited 2-Stage Diagonally Implicit Runge Kutta

(L-DIRK2) methods [4]. Numerical results showed that applying time-limiters eﬀectively reduces the numerical

oscillations when the time step is beyond the SSP limits of the original linear time integration schemes, and the

second order accuracy is still maintained.

In modern applications such as turbulence simulations, very high order spatial reconstructions, such as the 5th

order WENO scheme, may be applied. Third order (or even higher order) implicit time integration would better

preserve the high order of accuracy. An appealing option is the DIRK3 scheme, which, however, can be highly

oscillatory under large CFL numbers. In the following discussions we will apply the idea of time limiting to improve

the performance DIRK3.

3 Brief Review of the DIRK3 Scheme

In this section, we brieﬂy review the properties of the DIRK3 scheme and explain the motivation of applying

time-limiters. For convenience of notation, we represent the set of ODEs (2.2) as a time-depedent system

ut=L(u),u(t) = [u1(t), u2(t),· · · , uN(t)]>.(3.1)

The right hand side L(u) represents the spatial discretization of the numerical scheme. The DIRK3 time integration

scheme can be represented by the Butcher array (R. Alexander [1])

α α 0 0

τ2τ2−α α 0

1b1b2α

b1b2α

,(3.2)

where α≈0.435866521508459 is the root of x3−3x2+3

2x−1

6= 0 lying in ( 1

6,1

2),

τ2=1 + α

b1=−6α2−16α+ 1

b3=6α2−20α+ 5

It was shown [1] that (3.2) is the unique strongly S-stable DIRK formula of order three in three stages. The

S-stability of RK methods is deﬁned as follows.

S-stability ([14]). A RK method is S-stable if for any bounded g: [0, T ]7→ Rhaving a bounded derivative, and

any positive constant λ0, there is a positive constant h0such that the numerical solution {yn}, computed with

time step h, to the equation

y0=g0(t) + λ(y−g(t))

satisﬁes

|yn+1 −g(tn+1)

yn−g(tn)|<1

provided yn6=g(tn) for all 0 < h < h0all complex λwith Re(−λ)≥λ0.

A RK method is strongly S-stable if

yn+1 −g(tn+1)

yn−g(tn)→0

as Re(−λ)→ ∞ for all h > 0 such that [tn, tn+1]⊂[0, T ].

We notice that S-stability implies A-stability, which can be recovered by taking g≡0. Being S-stable guarantees

the stability of a method when applied to problems of stiﬀ equations. However, this does not ensure the solutions

to be non-oscillatory. In fact, the oscillatory nature of DIRK3 can be expected. We look at the ﬁrst two stages of

DIRK3

stage 1:

u(1) =un+α∆tL(u(1)).

stage 2:

u(2) =un+ (τ2−α)∆tL(u(1)) + α∆tL(u(2))

=u(1) + (τ2−2α)∆tL(u(1)) + α∆tL(u(2)).

Here the superscript of urepresents the number of stage. We notice that the second stage includes a backward

time stepping of u(1) (notice that τ2−2α < 0), which is unconditionally not TVD and may lead to oscillations in

the solution. Such instability motivates the introduction of limiting mechanisms in the regions where the solution

has large gradients.

4 The Limited-DIRK3 Scheme

4.1 One-dimensional constructions

We construct the L-DIRK3 scheme for conservation laws and convection - diﬀusion equations. Denote τ:= ∆t

∆x.

The L-DIRK3 scheme for conservation equation (2.1) is given by

u(1)

j=un

j−τα(ˆ

f(1)

j+1

−ˆ

f(1)

j−1

)

u(2)

j=un

j−τ[(a21,j+1

f(1)

j+1

−a21,j−1

f(1)

j−1

)+(a22,j+1

f(2)

j+1

−a22,j−1

f(2)

j−1

)]

un+1

j=un

j−τ[(a31,j+1

f(1)

j+1

−a31,j−1

f(1)

j−1

)+(a32,j+1

f(2)

j+1

−a32,j−1

f(2)

j−1

)

+ (a33,j+1

fn+1

j+1

−a33,j−1

fn+1

j−1

)]

,(4.1)

where

a21,j+1

2=α+θ(1)

j+1

(τ2−2α)

a22,j+1

2=1−α

2+θ(1)

j+1

(3α−1

a31,j+1

2=α+θ(2)

j+1

(b1−α)

a32,j+1

2=1−α

2+θ(2)

j+1

(b2−1−α

a33,j+1

2=1−α

2+θ(2)

j+1

(3α−1

, θ(k)

j+1

=θ(k)

j+θ(k)

j+1

2, θ(k)

j∈[0,1].

The values of α,τ2,b1,b2are as deﬁned in section 3. The time-limiter, θ(k)

j, measures the local smoothness of

solution at stage k. It is clear that the scheme (4.1) is conservative and consistent.

To illustrate the idea of construction, we ﬁrst keep θ(k)

j≡θas constant at all points in the domain and at all

stages. The resulting scheme reads

α α 0 0

τ2α+θ(τ2−2α)1−α

2+θ(3α−1

2) 0

1α+θ(b1−α)1−α

2+θ(b2−1−α

2)1−α

2+θ(3α−1

α+θ(b1−α)1−α

2+θ(b2−1−α

2)1−α

2+θ(3α−1

.(4.2)

When θ= 1 we recover the DIRK3 scheme (3.2), when θ= 0 we obtain the unconditionally SSP successive implicit

Euler steps (IE-IE-IE),

α α 0 0

τ2α1−α

1α1−α

1−α

α1−α

1−α

,(4.3)

or equivalently

u(1) =un+α∆tL(u(1)),

u(2) =u(1) + (τ2−α)∆tL(u(2))

un+1 =u(3) =u(2) + (1 −τ2)∆tL(u(3))

Following the idea of [4], it is expected that the ﬁrst order method (4.3) is applied locally near discontinuities and

extrema, whereas in the smooth regions we still apply the third order accurate DIRK3. Hence, the local θ(k)

jshould

be applied.

Here we propose a new and convenient construction for θ(k)

j. For scalar problems, we deﬁne

θ(k)

j=minmod(r(k)

j,1),

r(k)

j=u(k+1)

j+1 −u(k+1)

j−1

u(k)

j+1 −u(k)

j−1

, k = 1,2, j = 1,2,· · · N. (4.4)

The indicator function r(k)

j≈∂xu(k+1)

∂xu(k)

is used to detect the change in the solution monotonicity at point xjat

stage k. In the smooth and monotone regions we expect r(k)

j≈1, then we would have θ(k)

j≈1 and recover the

DIRK3 scheme. If the solution changes the monotonicity at point xjwhen proceeds from stage kto stage k+ 1

(i.e. ∂xu(k)

j·∂xu(k+1)

j≤0), which usually implies the occurrence of discontinuity or extrema, r(k)

jwould be closed

to 0 or become negative, then we would apply θ(k)

j≈0 and recover successive implicit Euler steps (4.3).

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Non-OcillatoryLimited-TimeIntegrationforConservationLawsandConvection-DiusionEquationsJingchengLuandJamesD.BaederAbstractInthisstudyweconsiderunconditionallynon-oscillatory,highorderimplicittimemarchingbasedontime-limiters.TherstaspectofourworkistoproposethehighresolutionLimited-DIRK3(L-DIRK3)sche...

展开>> 收起<<

Non-Ocillatory Limited-Time Integration for Conservation Laws and Convection-Diusion Equations Jingcheng Lu and James D.Baeder.pdf

共17页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Non-Ocillatory Limited-Time Integration for Conservation Laws and Convection-Diusion Equations Jingcheng Lu and James D.Baeder

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: