General Formulation of Second-Order Semi-Lagrangian Methods for Convection-Diffusion Problems

Abstract

The general formulation of the second-order semi-Lagrangian methods was presented for convection-dominated diffusion problems. In view of the method of lines, this formulation is in a sufficiently general fashion as to include two-step backward difference formula and Crank-Nicolson type semi-Lagrangian schemes as particular ones. And it is easy to be extended to higher-order schemes. We show that it maintains second-order accuracy even if the involved numerical characteristic lines are first-order accurate. The relationship between semi-Lagrangian methods and the modified method of characteristic is also addressed. Finally convergence properties of the semi-Lagrangian finite difference schemes are tested.

1. Introduction

There have been many numerical methods developed to deal with convection-dominated diffusion problems, and among them characteristic-based methods are the most popular. Numerical methods that follow characteristic lines backwards in time and then interpolate at their feet date back to the work of Courant et al. [1]. As one kind of characteristics-based method, the semi-Lagrangian (SL) method was introduced in the beginning of the 1980s by Robert [2]. Its basic idea is to discretize the Lagrangian derivative of the solution instead of the Eulerian derivative. This technique can increase significantly the maximum allowable time step while maintaining the efficiency of symmetric solvers. The SL methods have been extensively applied in numerical simulations of models for weather forecast and oceanography (see Staniforth and Côté [3] and Smolarkiewicz and Pudykiewicz [4] for review). As another kind of characteristic-based methods, the modified method of characteristic (MMOC) was introduced by Douglas Jr. and Russell [5] at roughly the same time as the SL method and has been extensively implemented in numerical simulations of fluid flows in porous media (see [6–8]) and many other transport problems (see [9–12] for review).

Strang [13] pointed out that the first-order methods were often too crude and the third-order methods too complicated. The computations are thus made expensive either by the fine mesh required by a first order scheme in order to provide enough detail or else by the delicate differencing which maintains a high-order accuracy. Second-order schemes are the obvious compromise.

Based on MMOC, in the beginning of 1980s Ewing and Russell [14] introduced the backward difference formula (BDF) of characteristic for linear constant-coefficient convection-diffusion problems. Afterwards characteristic schemes of Crank-Nicolson (CN) type for convection-diffusion equations and the Navier-Stokes (NS) equations were studied by Rui and Tabata [15] and Notsu and Tabata [16], respectively. The SL-BDF schemes for incompressible NS problems were proposed by Boukir et al. [17, 18]. The SL-CN methods were proposed for convection-diffusion equations and/or the incompressible NS equations by Bermúdez et al. [19, 20], Fourestey and Piperno [21], Xiu and Karniadakis [22], and Xiu et al. [23]. Al-Lawatia et al. [24] and Falcone and Ferretti [25] presented and analyzed the single-step high-order semi-Lagrangian schemes of the Runge-Kutta type. More research was given by Bermejo and Conde [26], Xiao and Yabe [27], and Toda et al. [28].

Up to now we have not seen the SL method to be formulated in a sufficiently general fashion. In this paper we present a general second-order SL formulation for convection-dominated diffusion problems. By the method of lines approach, this formulation includes most of the second-order SL schemes mentioned previously. Extension of it to higher-order schemes is also addressed. Our development of the general SL formulation is mainly motivated by the work of Rui and Tabata [15] and Notsu and Tabata [16]. They treated convection-diffusion and the NS problem, respectively, by MMOC. They emphasized that an “additional correction term” was indispensable to maintain the second-order accuracy of the MMOC schemes. We will see that the correction term of the MMOC schemes is actually a natural term of the SL schemes.

First, we show that by the method of lines (MOL) approach, the general SL formulation includes the SL-BDF2 and the SL-CN schemes as specific ones. Second, we verify that the formulation maintains second-order accuracy even if the involved numerical characteristic lines are first-order accurate. Third, we show that MMOC can be considered as a special version of the SL method. Finally, combining finite difference discretization in spaces, a fully discretized SL scheme is presented. Numerical tests demonstrate that the SL finite difference scheme is second-order convergent.

The outline of the rest of this paper is as follows. In Section 2 the forming of the SL schemes is recalled and the relationship between the first-order SL methods and MMOC is addressed. In Section 3 the general SL formulation is derived and its second-order consistency is proved. The relation of the second-order SL method and MMOC is also addressed. In Section 4 the finite difference SL schemes are derived. In Section 5 several SL finite difference schemes are applied to the Gaussian hill problem. In Section 6 conclusions of this paper are drawn.

2. The SL Methods and the MMOC

In this section we recall some details of the construction of the SL method and MMOC. Also we study the relationship between them.

Let Ω be a bounded domain in ℝ² with Lipschitz boundary 𝜕Ω, and let 𝑇 be a positive constant. Without loss of generality, we consider the initial boundary value problem: find 𝜙∶Ω×(0,𝑇)→ℝ such that𝜕𝜙(),𝜙()(),𝜙()𝜕𝑡+𝐮⋅∇𝜙−𝜈Δ𝜙=𝑓inΩ×0,𝑇𝐱,𝑡=0on𝜕Ω×0,𝑇𝐱,0=𝜙₀(𝐱)inΩ, (1a)(1b)(1c)where 𝜈 is a positive constant, 𝐮∶Ω×(0,𝑇)→ℝ², and 𝑓∶Ω×(0,𝑇)→ℝ are given functions. For simplicity, we assume that 𝐮 is a divergence-free velocity field, that is, ∇⋅𝐮=0, and vanishes on the boundary 𝜕Ω.

Characteristic line is the trajectory of a fluid particle. The travel of the particle is associated with the velocity field 𝐮. For a given point (𝐱,𝑠)∈Ω×[0,𝑇], the characteristic line through (𝐱,𝑠) is represented by the vector function 𝐗(𝐱,𝑠;𝑡), which solves the initial value problemd𝐗(𝐱,𝑠;𝑡)(𝐗()),𝐗()d𝑡=𝐮𝐱,𝑠;𝑡,𝑡𝐱,𝑠;𝑠=𝐱. (2a)(2b)Here 𝐗(𝐱,𝑠;𝑡) is the position of the particle on the characteristic line at time 𝑡. The particle is located at 𝐱 at time 𝑠. We assume that 𝐮∈𝐶⁰(Ω×[0,𝑇]) is Lipschitz continuous on 𝜕Ω with respect to the first variable. By the theory of ODEs, the characteristic line is well defined. By the chain rule we have d𝜙(𝐗())=d𝑡𝐱,𝑠;𝑡,𝑡𝜕𝜙(𝐗())(𝐗())()(),𝜕𝑡𝐱,𝑠;𝑡,𝑡+𝐮𝐱,𝑠;𝑡,𝑡⋅∇𝜙⋅,𝑡∘𝐗𝐱,𝑠;𝑡 (3) where 𝜙(⋅,𝑡)∘𝐗(𝐱,𝑠;𝑡) is the composition of functions with respect to the first argument of 𝜙. Similar to the deriving of (4.1) and (4.2) in [19], (1a) can be written to Lagrangian form d𝜙(𝐗())()()(𝐗()).d𝑡𝐱,𝑠;𝑡,𝑡−𝜈Δ𝜙⋅,𝑡∘𝐗𝐱,𝑠;𝑡=𝑓𝐱,𝑠;𝑡,𝑡 (4)

For a positive integer 𝑁, let Δ𝑡=𝑇/𝑁 be time step length and 𝑡_𝑛=𝑛Δ𝑡, for 𝑛=1,2,…,𝑁. Let 𝐗(𝐱,𝑡_𝑛+1;𝑡) denote the characteristic line on [𝑡_𝑛,𝑡_𝑛+1] (or [𝑡_𝑛−1,𝑡_𝑛+1] for two-step methods) through (𝐱,𝑡_𝑛+1). Thus (4) can locally be written to d𝜙𝐗d𝑡𝐱,𝑡_𝑛+1;𝑡,𝑡−𝜈Δ𝜙(⋅,𝑡)∘𝐗𝐱,𝑡_𝑛+1𝐗;𝑡=𝑓𝐱,𝑡_𝑛+1;𝑡,𝑡. (5)

Tracking the particle backward from 𝐱 to 𝐗(𝐱,𝑡_𝑛+1;𝑡_𝑛) along the characteristic line on [𝑡_𝑛,𝑡_𝑛+1] (or from 𝐱 to 𝐗(𝐱,𝑡_𝑛+1;𝑡_𝑛−1) on [𝑡_𝑛−1,𝑡_𝑛+1] for two-step methods) corresponds to the backward solution of the following Cauchy problem: d𝐗𝐱,𝑡_𝑛+1;𝑡𝐗d𝑡=𝐮𝐱,𝑡_𝑛+1,𝐗;𝑡,𝑡𝐱,𝑡_𝑛+1;𝑡_𝑛+1=𝐱. (6) For function 𝑤∶Ω×[𝑡_𝑛,𝑡_𝑛+1]→ℝ, denote 𝑤^𝑖=𝑤(𝐱,𝑡_𝑖) and 𝐗^𝑖=𝐗(𝐱,𝑡_𝑛+1,𝑡_𝑖) for 𝑖=𝑛,𝑛+1. By applying the backward Euler’s method to (5), it follows that 𝜑^𝑛+1−𝜑^𝑛𝐗^𝑛Δ𝑡−𝜈Δ𝜑^𝑛+1=𝑓^𝑛+1, (7) where 𝜑^𝑛 represents the approximate of 𝜙^𝑛. Note that in practice, (6) can usually be solved approximately. By applying the backward Euler’s method to (6), it follows that 𝐱−𝐗^𝑛_𝐸Δ𝑡=𝐮^𝑛+1(𝐱), (8) and 𝐗^𝑛_𝐸=𝐱−𝐮^𝑛+1Δ𝑡 (similarly we see that 𝐗_𝐸^𝑛−1=𝐱−2𝐮^𝑛+1Δ𝑡 for the two-step methods). In (7) with 𝐗^𝑛 being replaced by 𝐗^𝑛_𝐸, we have 𝜑^𝑛+1−𝜑^𝑛𝐱−𝐮^𝑛+1Δ𝑡Δ𝑡−𝜈Δ𝜑^𝑛+1=𝑓^𝑛+1. (9)

On the other hand, (9) can be derived by MMOC [5]. In fact, with 𝐮=(𝑢₁,𝑢₂), let 𝐬 denote the direction vector (1,𝑢₁,𝑢₂), and define the operator d1d𝐬∶=𝜃𝜕𝜕𝑡+𝐮⋅∇, (10) with 𝜃(𝐱,𝑡)∶=[1+|𝐮(𝐱,𝑡)|²]^1/2=[1+|𝑢₁(𝐱,𝑡)|²+|𝑢₂(𝐱,𝑡)|²]^1/2. So (1a) can be written to the form 𝜃d𝜙d𝐬−𝜈Δ𝜙=𝑓. (11) Using the backward difference quotient, we have d𝜙^𝑛+1=𝜙d𝐬^𝑛+1−𝜙^𝑛𝐱−Δ𝑡𝐮^𝑛+1/𝜃^𝑛+1+Δ𝑡Δ𝑡2d²𝜙^𝑛+1d²𝐬+𝑂Δ𝑡². (12) Substituting (12) into (11), we obtain (9). Thus MMOC can be considered as a special version of SL. In the next section we will see that similar relation exists for second-order case.

3. General Second-Order SL Formulation

In this section we present a general second-order SL formulation and show that by MOL this formulation includes several widely used schemes as specific ones.

Let us first consider the abstract ODEs of the following form: given the Hilbert space ℋ and 𝑦₀∈ℋ, find 𝑦∶(0,𝑇]→ℋ such that 𝑦^′(𝑡)(𝑡)(],𝑦(0)=𝑔𝑡,𝑦,𝑡∈0,𝑇=𝑦₀, (13) where 𝑔∶(0,𝑇]×ℋ→ℋ. A general second-order scheme for (13) can be of the form (see [29, 30]) ²_ℓ=0𝛼_ℓ𝑌^𝑛+ℓ−1=Δ𝑡²_ℓ=0𝛽_ℓ𝑔^𝑛+ℓ−1𝑌^𝑛+ℓ−1,𝑛=1,2,…,𝑁−1. (14) As usual we denote scheme (14) as (𝜌,𝜎), where 𝜌 and 𝜎 are the characteristic polynomials of the scheme, with 𝜌(𝜁)∶=²_ℓ=0𝛼_ℓ𝜁^ℓ,𝜎(𝜁)∶=²_ℓ=0𝛽_ℓ𝜁^ℓ. (15) We further assume that (𝜌,𝜎) is normalized by ∑²_ℓ=0𝛽_ℓ=1 and satisfies the following conditions: 𝛼₁=1−2𝛼₂,𝛼₀=−1+𝛼₂,𝛽₁=12+𝛼₂−2𝛽₂,𝛽₀=12−𝛼₂+𝛽₂, (16) for 𝛼₂≥1/2, 𝛽₂≥𝛼₂/2, such that (𝜌,𝜎) is Dahlquist and 𝐴-stable [31].

For a fixed 𝐱∈Ω and 𝑡∈[𝑡_𝑛−1,𝑡_𝑛+1], let 𝑦(𝑡)=𝜙(𝐗(𝐱,𝑡_𝑛+1;𝑡),𝑡). Then the (𝜌,𝜎) scheme for (5) is of the form ²_ℓ=0𝛼_ℓ𝜑^𝑛+ℓ−1𝐗^𝑛+ℓ−1=Δ𝑡²_ℓ=0𝛽_ℓ𝜈Δ𝜑^𝑛+ℓ−1∘𝐗^𝑛+ℓ−1+𝑓^𝑛+ℓ−1𝐗^𝑛+ℓ−1,𝑛=1,2,…,𝑁−1. (17) With 𝐗 being replaced by the approximate characteristic line 𝐗, we have the analogue of (17): ²_ℓ=0𝛼_ℓ𝜑^𝑛+ℓ−1𝐗^𝑛+ℓ−1=Δ𝑡²_ℓ=0𝛽_ℓ𝜈Δ𝜑^𝑛+ℓ−1∘𝐗^𝑛+ℓ−1+𝑓^𝑛+ℓ−1𝐗^𝑛+ℓ−1,𝑛=1,2,…,𝑁−1. (18) Scheme (18) is the general second-order SL formulation which by MOL includes all the previously introduced second-order SL schemes. Next we will prove that it includes the SL-BDF2 schemes and the SL-CN schemes as specific cases.

3.1. The BDF2 Scheme

In (14), let 𝛽₀=𝛽₁=0; then from (16), we have 𝛼₂=3/2, 𝛼₁=−2, 𝛼₀=1/2. Substituting these coefficients into (14), we obtain the BDF2 scheme: 3𝑌^𝑛+1−4𝑌^𝑛+𝑌^𝑛−12Δ𝑡=𝑔^𝑛+1𝑌^𝑛+1. (19) Due to its stability and damping properties, (19) is one of the most popular second-order schemes [29]. Substituting the previously coefficients into (17), we get the SL analogue of (19): 3𝜑^𝑛+1−4𝜑^𝑛𝐗^𝑛+𝜑^𝑛−1𝐗^𝑛−12Δ𝑡−𝜈Δ𝜑^𝑛+1=𝑓^𝑛+1. (20) Furthermore, in (20) with 𝐗^𝑛 and 𝐗^𝑛−1 being, respectively, approximated by 𝐗^𝑛_𝐸=𝐱−𝐮^𝑛+1Δ𝑡, 𝐗_𝐸^𝑛−1=𝐱−2𝐮^𝑛+1Δ𝑡, we have 3𝜑^𝑛+1−4𝜑^𝑛𝐱−𝐮^𝑛+1Δ𝑡+𝜑^𝑛−1𝐱−2𝐮^𝑛+1Δ𝑡2Δ𝑡−𝜈Δ𝜑^𝑛+1=𝑓^𝑛+1. (21) This is just the multistep characteristic scheme derived based on MMOC in [14]. In fact, using MMOC (analogous to (12)) we see that d𝜙^𝑛+1=d𝐬3𝜙^𝑛+1−4𝜙^𝑛𝐱−Δ𝑡𝐮^𝑛+1/𝜃^𝑛+1+𝜙^𝑛−1𝐱−2Δ𝑡𝐮^𝑛+1/𝜃^𝑛+1(2Δ𝑡)⁻¹+(Δ𝑡)²3d³𝜙^𝑛+1d𝐬³+𝑂Δ𝑡³. (22) Substituting (22) into (11), we obtain (21).

The SL-BDF2 schemes were presented and analyzed in [14, 17, 18, 21]. It is deserved to note that though the involved approximate characteristic line is first-order accurate, the resulting SL-BDF2 scheme (21) maintains second-order accurate.

3.2. The CN Scheme

In (14), let 𝛼₂=1,𝛼₁=−1,𝛼₀𝛽=0;₂=12,𝛽₁=𝛽₀=0. (23) From (14) we obtain the CN scheme 𝑌^𝑛+1−𝑌^𝑛=1Δ𝑡2𝑔^𝑛𝑌^𝑛+𝑔^𝑛+1𝑌^𝑛+1. (24) From (17) we get the SL analogue of (24): 𝜑^𝑛+1−𝜑^𝑛𝐗^𝑛−1Δ𝑡2𝜈Δ𝜑^𝑛∘𝐗^𝑛+𝜑^𝑛+1=12𝑓^𝑛+1+𝑓^𝑛𝐗^𝑛, (25) and with 𝐗^𝑛 being replaced by 𝐗_𝐸, we have 𝜑^𝑛+1−𝜑^𝑛𝐱−𝐮^𝑛+1Δ𝑡−1Δ𝑡2𝜈Δ𝜑^𝑛∘𝐱−𝐮^𝑛+1Δ𝑡+𝜑^𝑛+1=12𝑓^𝑛𝐱−𝐮^𝑛+1Δ𝑡+𝑓^𝑛+1. (26)

Remark 1. It is deserved to note that Rui and Tabata [15] and Notsu and Tabata [16] called (1/2)𝜈Δ𝜑^𝑛∘(𝐱−𝐮^𝑛+1Δ𝑡)-term in (26) the “additional corrected term,” since it is introduced from “outside” to recover the second-order consistency of the MMOC schemes. But in view of the previous discussion, the “additional corrected term” is a natural one in the SL schemes. Thus we think the SL method is more general than MMOC.

3.3. The Consistency of the General Formulation

Now we show that both (17) and (18) with first-order approximate characteristic lines are second-order accurate. Let us denote ℒ[𝑤(𝑡);Δ𝑡]∶=²_ℓ=0𝛼_ℓ𝑤^𝑛+ℓ−1−Δ𝑡𝛽_ℓ𝑔^𝑛+ℓ−1𝑤^𝑛+ℓ−1, (27) for 𝑤∶(0,𝑇]→ℋ. By the theory of ODEs (see [30]), using the Taylor’s expansion, from (13) and (14) we have ℒ𝑦(𝑡)=𝛼;Δ𝑡₀+𝛼₁+𝛼₂𝑦(𝑡)+Δ𝑡𝛼₁+2𝛼₂−𝛽₀+𝛽₁+𝛽₂𝑦^′(𝑡)+Δ𝑡²12𝛼₁+4𝛼₂−𝛽₁+2𝛽₂𝑦^′′(𝑡)+𝑂Δ𝑡³. (28) If the following conditions hold 𝛼₀+𝛼₁+𝛼₂𝛼=0,₁+2𝛼₂−𝛽₀+𝛽₁+𝛽₂1=0,2𝛼₁+4𝛼₂−𝛽₁+2𝛽₂=0, (29) then ℒ[𝑦(𝑡);Δ𝑡]=𝑂(Δ𝑡³). It is easy to see that conditions (29) and (16) are equivalent if ∑²_ℓ=0𝛽_ℓ=1. In (13), let 𝑦(𝑡)=𝜑𝐗⋅,𝑡_𝑛+1,𝑔𝑦(𝑡)();𝑡,𝑡,𝑡=𝜈Δ𝜑⋅,𝑡∘𝐗⋅,𝑡_𝑛+1;𝑡+𝑓𝐗⋅,𝑡_𝑛+1;𝑡,𝑡. (30) From (28)–(30) it follows that ℒ𝜙𝐗⋅,𝑡_𝑛+1;𝑡,𝑡;Δ𝑡=𝑂Δ𝑡³, (31) where 𝜙 is the exact solution of (5). Thus we have confirmed the following proposition.

Proposition 2. Suppose that 𝐮 and 𝑓 in (1a)–(1c) are smooth functions in Ω×(0,𝑇), and 𝐗(𝑡) is the exact characteristic line which solves (6). Then (17) is second-order consistent with (5).

Analogous to Proposition 2, if first-order characteristic lines are involved, then the following proposition holds.

Proposition 3. Suppose that 𝐮 and 𝑓 in (1a)–(1c) are smooth functions in Ω×(0,𝑇). In (18), if 𝐗^𝑛=𝐗^𝑛_𝐸, 𝐗^𝑛−1=𝐗_𝐸^𝑛−1, then the scheme ²_ℓ=0𝛼_ℓ𝜑^𝑛+ℓ−1𝐗_𝐸^𝑛+ℓ−1−𝜈Δ𝑡²_ℓ=0𝛽_ℓΔ𝜑^𝑛+ℓ−1∘𝐗_𝐸^𝑛+ℓ−1=Δ𝑡²_ℓ=0𝛽_ℓ𝑓^𝑛+ℓ−1𝐗_𝐸^𝑛+ℓ−1 (32) is second-order consistent with (5).

Proof. Analogous to the deriving of (4.1) and (4.2) in [19], for 𝑡∈[𝑡_𝑛−1,𝑡_𝑛+1] (5) can be written to the form 𝜕𝜙(⋅,𝑡)𝜕𝑡∘𝐗𝐱,𝑡_𝑛+1;𝑡+𝐮𝐗𝐱,𝑡_𝑛+1();𝑡,𝑡⋅∇𝜙⋅,𝑡∘𝐗𝐱,𝑡_𝑛+1;𝑡−𝜈Δ𝜙(⋅,𝑡)∘𝐗𝐱,𝑡_𝑛+1𝐗;𝑡=𝑓𝐱,𝑡_𝑛+1;𝑡,𝑡. (33) Noting that 𝐗𝐱,𝑡_𝑛+1;𝑡=𝐱−Δ𝑡𝐮^𝑛+1(𝐱)+𝑂Δ𝑡²,𝐗𝐱,𝑡_𝑛+1;𝑡=𝐱−2Δ𝑡𝐮^𝑛+1(𝐱)+𝑂Δ𝑡², (34) we denote 𝐗_𝐸𝐱,𝑡_𝑛+1;𝑡=𝐱−Δ𝑡𝐮^𝑛+1(𝐱),𝐗_𝐸𝐱,𝑡_𝑛+1;𝑡=𝐱−2Δ𝑡𝐮^𝑛+1(𝐱). (35) With 𝐗(𝐱,𝑡_𝑛+1;𝑡) being replaced by 𝐗_𝐸(𝐱,𝑡_𝑛+1;𝑡) and 𝜑 being the approximate of 𝜙, we obtain 𝑂(Δ𝑡²)-perturbation of (33) of the form 𝜕𝜑(⋅,𝑡)𝜕𝑡∘𝐗_𝐸𝐱,𝑡_𝑛+1;𝑡+𝐮𝐗_𝐸𝐱,𝑡_𝑛+1();𝑡,𝑡⋅∇𝜑⋅,𝑡∘𝐗𝐱,𝑡_𝑛+1;𝑡−𝜈Δ𝜑(⋅,𝑡)∘𝐗_𝐸𝐱,𝑡_𝑛+1𝐗;𝑡=𝑓_𝐸𝐱,𝑡_𝑛+1;𝑡,𝑡. (36) Rewrite (36) to the Lagrangian form d𝜑(⋅,𝑡)𝐗d𝑡^𝐸𝐱,𝑡_𝑛+1;𝑡,𝑡−𝜈Δ𝜑(⋅,𝑡)∘𝐗_𝐸𝐱,𝑡_𝑛+1𝐗;𝑡=𝑓_𝐸𝐱,𝑡_𝑛+1;𝑡,𝑡. (37) Thus (37) is the second-order approximation to (5).
Corresponding to (30), let 𝑦(𝑡)=𝜑𝐗_𝐸⋅,𝑡_𝑛+1,𝑔𝑦(𝑡)();𝑡,𝑡,𝑡=𝜈Δ𝜑⋅,𝑡∘𝐗_𝐸𝐱,𝑡_𝑛+1;𝑡+𝑓𝐗_𝐸⋅,𝑡_𝑛+1;𝑡,𝑡. (38) Similar to the proof of Proposition 2, we can see that (32) is second-order consistent with (37). We deduce that (32) is second-order consistent with (5).

Remark 4. In (32) higher-order numerical characteristic lines are usually preferred, though the characteristic line computed by the backward Euler’s method can retain the second-order accuracy. For example, Bermúdez et al. [19, 20], Rui and Tabata [15], and Notsu and Tabata [16] computed the numerical characteristic by the higher-order Runge-Kutta methods. In the following numerical tests we will see that the first-order characteristic line is too coarse to ensure reasonable convergence of the SL-CN schemes.

4. The SL Finite Difference Method

In this section we present a full-discretized SL formulation which combines finite difference for spatial discretizations. We also numerically verify the convergence of the formulation.

First, we build the finite difference scheme for (5). Assume that Ω is a unit square, with boundary 𝜕Ω. Denote 𝜕Ω=Γ₁∪Γ₂∪Γ₃∪Γ₄, with Γ₁,Γ∶=𝑥,𝑦∣0≤𝑥≤1,𝑦=0₂,Γ∶=𝑥,𝑦∣𝑥=1,0≤𝑦≤1₃,Γ∶=𝑥,𝑦∣0≤𝑥≤1,𝑦=1₄∶=𝑥,𝑦∣𝑥=0,0≤𝑦≤1. (39) For the partition of Ω, we denote ℎ∶=𝑀⁻¹,𝐱_𝑖𝑗∶=(𝑖ℎ,𝑗ℎ) and 𝑤(𝐱_𝑖𝑗,𝑡)∶=𝑤_𝑖𝑗(𝑡). Let Ω_ℎ∶=𝜔₁×𝜔₂, with 𝜔₁∶=𝑥_𝑖∣0≤𝑖≤𝑀,𝜔₂∶=𝑦_𝑗∣0≤𝑗≤𝑀. (40) Let Ω_ℎ∶=𝜔₁×𝜔₂, with 𝜔₁∶=𝑥_𝑖∣1≤𝑖≤𝑀−1,𝜔₂∶=𝑦_𝑗∣1≤𝑗≤𝑀−1. (41) Let 𝐱∶=(𝑥,𝑦), 𝐗∶=(𝑋,𝑌) with 𝐗(𝑥,𝑦,𝑡)=(𝑋(𝑥,𝑦,𝑡),𝑌(𝑥,𝑦,𝑡)). By the transformation given in Appendix section, term Δ𝜙(⋅,𝑡)∘𝐗(𝐱,𝑡_𝑛+1;𝑡) in (5) changes to expression that consists of 𝜕²𝜙/𝜕𝑥², 𝜕²𝜙/𝜕𝑦², 𝜕²𝜙/𝜕𝑥𝜕𝑦, 𝜕𝜙/𝜕𝑥, 𝜕𝜙/𝜕𝑦, 𝜕²𝑋/𝜕𝑥², 𝜕²𝑋/𝜕𝑦², 𝜕²𝑋/𝜕𝑥𝜕𝑦, 𝜕𝑋/𝜕𝑥, 𝜕𝑋/𝜕𝑦, and so forth. If 𝑋(𝑥,𝑦,𝑡), 𝑌(𝑥,𝑦,𝑡) are polynomials of 𝑥,𝑦 of degrees not more than one, then it holds that (see Appendix) Δ𝜙(⋅,𝑡)∘𝐗𝐱,𝑡_𝑛+1=𝜕;𝑡₂𝜙𝜕𝑥²⋅𝜕𝑋𝜕𝑦²+𝜕𝑌𝜕𝑦²+𝜕²𝜙𝜕𝑦²⋅𝜕𝑋𝜕𝑥²+𝜕𝑌𝜕𝑥²−2𝜕²𝜙𝜕𝑥𝜕𝑦𝜕𝑋⋅𝜕𝑥𝜕𝑋+𝜕𝑦𝜕𝑌⋅𝜕𝑥𝜕𝑌⋅𝜕𝑦𝜕𝑌⋅𝜕𝑥𝜕𝑋−𝜕𝑦𝜕𝑌⋅𝜕𝑦𝜕𝑋𝜕𝑥⁻². (42) Using centered difference to discretize these partial derivatives, we obtain the finite difference approximation of (5) as follows: find 𝜙_ℎ∶Ω_ℎ×(0,𝑇)→ℝ such that d𝜙_ℎ𝐗𝐱d𝑡^𝑖𝑗,𝑡_𝑛+1;𝑡,𝑡−𝜈Δ_ℎ𝜙(⋅,𝑡)∘𝐗𝐱_𝑖𝑗,𝑡_𝑛+1𝐗𝐱;𝑡=𝑓_𝑖𝑗,𝑡_𝑛+1;𝑡,𝑡, (43) where Δ_ℎ𝜙(⋅,𝑡)∘𝐗(𝐱_𝑖𝑗,𝑡_𝑛+1;𝑡) is the approximate of Δ𝜙(⋅,𝑡)∘𝐗(𝐱,𝑡_𝑛+1;𝑡). Applying (17) to (43), with Φ representing the approximate of 𝜙_ℎ, we obtain the second-order-in-time finite difference scheme:²_𝑙=0𝛼_𝑙Φ^𝑛+ℓ−1𝐗^𝑛+ℓ−1_𝑖𝑗−𝜈Δ𝑡Δ_ℎ²_𝑙=0𝛽_𝑙Φ^𝑛+ℓ−1∘𝐗^𝑛+ℓ−1_𝑖𝑗=Δ𝑡²_𝑙=0𝛽_𝑙𝑓^𝑛+ℓ−1𝐗^𝑛+ℓ−1_𝑖𝑗,𝑖,𝑗=1,2,…,𝑀−1,𝑛=1,2,…,𝑁−1. (44) In (44) with 𝐗 being replaced by 𝐗_𝐸, Φ representing the approximate of Φ, we finally discretize (1a)–(1c) as follows: ²_𝑙=0𝛼_𝑙Φ^𝑛+ℓ−1_𝑖𝑗𝐗_𝐸^𝑛+ℓ−1−𝜈Δ𝑡Δ_ℎ²_𝑙=0𝛽_𝑙Φ^𝑛+ℓ−1_𝑖𝑗∘𝐗_𝐸^𝑛+ℓ−1=Δ𝑡²_𝑙=0𝛽_𝑙𝑓^𝑛+ℓ−1_𝑖𝑗𝐗_𝐸^𝑛+ℓ−1𝑖,𝑗=1,2,…,𝑀−1,𝑛=1,2,…,𝑁−1. (45) The computation of the starting values of this multistep scheme is similar to the general Eulerian schemes. Since 𝐗^𝑛_𝐸 and 𝐗_𝐸^𝑛−1 in (45) are not grid points of Ω_ℎ, interpolations are needed. In the following numerical tests, we will use cubic interpolations (CI) and cubic spline interpolations, respectively (CSI).

5. Numerical Results

In this section, we test some specific cases of (45). The problem of rotating Gaussian hill has been widely used to test numerical schemes for convection-diffusion equations. In (1a)–(1c), let Ω=(−0.5,0.5)×(−0.5,0.5), 𝑇=𝜋/4, 𝐮=(−𝑦,𝑥), 𝑓=0, and 𝜈=1.25×10⁻⁴ or 𝜈=1.0×10⁻³. The Dirichlet boundary conditions and initial condition of (1a)–(1c) are given such that the exact solution is 𝑐=𝜎𝑥,𝑦,𝑡−𝜎+4𝜈𝑡exp𝑥(𝑡)−𝑥_𝑐²+𝑦(𝑡)−𝑦_𝑐²,𝜎+4𝜈𝑡 (46) where 𝑥(𝑡)=𝑥cos𝑡+𝑦sin𝑡, 𝑦(𝑡)=−𝑥sin𝑡+𝑦cos𝑡, (𝑥_𝑐,𝑦_𝑐)=(0.25,0) and 𝜎=0.01. The exact characteristic is given by 𝑋𝑌𝑥,𝑦,𝑡=𝑥cos𝑡−𝑦sin𝑡,𝑥,𝑦,𝑡=𝑥sin𝑡+𝑦cos𝑡. (47) By the Euler’s method the approximate characteristic is given by 𝑋^𝑛_𝐸𝑋𝑥,𝑦=𝑥+𝑦Δ𝑡,_𝐸𝑛−1𝑌𝑥,𝑦=𝑥+2𝑦Δ𝑡,_𝑛𝐸𝑌𝑥,𝑦=𝑦−𝑥Δ𝑡,_𝐸𝑛−1𝑥,𝑦=𝑦−2𝑥Δ𝑡. (48)

Note that since the velocity 𝐮 does not satisfy the nonflow boundary condition (1b), the characteristics and its approximations are not necessarily contained in Ω. We assume that 𝜙=0 outside of Ω (just as in Notsu and Tabata [16], Rui and Tabata [15], and Long and Yuan [32]).

Using the standard central difference along the numerical characteristic and the transformation given in Appendix, we have Δ_ℎΦ_𝑖𝑗∘𝐗_𝐸^𝑛+1=Δ_ℎΦ_𝑖𝑗=Φ^𝑛+1_𝑖+1,𝑗+Φ^𝑛+1_{𝑖−1,𝑗}+Φ^𝑛+1_𝑖,𝑗+1+Φ^𝑛+1_{𝑖,𝑗−1}−4Φ^𝑛+1_𝑖,𝑗ℎ²,Δ_ℎΦ_𝑖𝑗∘𝐗^𝑛_𝐸=Φ^𝑛_𝑖+1,𝑗+Φ^𝑛_{𝑖−1,𝑗}+Φ^𝑛_𝑖,𝑗+1+Φ^𝑛_{𝑖,𝑗−1}−4Φ^𝑛_𝑖,𝑗𝐗^𝑛_𝐸1+Δ𝑡²ℎ²,Δ_ℎΦ_𝑖𝑗∘𝐗_𝐸^𝑛−1=Φ^𝑛−1_𝑖+1,𝑗+Φ^𝑛−1_{𝑖−1,𝑗}+Φ^𝑛−1_𝑖,𝑗+1+Φ^𝑛−1_{𝑖,𝑗−1}−4Φ^𝑛−1_𝑖,𝑗𝐗_𝐸^𝑛−11+4Δ𝑡²ℎ². (49) Therefore the BDF2 scheme takes the form 3Φ^𝑛+1_𝑖𝑗−4Φ^𝑛_𝑖𝑗𝐗^𝑛_𝐸+Φ^𝑛−1_𝑖𝑗𝐗_𝐸^𝑛−1−2𝜈Δ𝑡Δ_ℎΦ^𝑛+1_𝑖𝑗=0, (50) and the CN scheme takes the form Φ^𝑛+1_𝑖𝑗−Φ^𝑛_𝑖𝑗𝐗^𝑛−𝜈Δ𝑡2Δ_ℎΦ^𝑛+1_𝑖𝑗−𝜈Δ𝑡2Δ_ℎΦ^𝑛_𝑖𝑗∘𝐗^𝑛_𝐸=0. (51) To compare the results, we need the first-step backward difference scheme (BDF1): Φ^𝑛+1_𝑖𝑗−Φ^𝑛_𝑖𝑗𝐗^𝑛_𝐸−𝜈Δ𝑡Δ_ℎΦ^𝑛+1_𝑖𝑗=0. (52) Let 𝐸_𝑙² denote the 𝑙²-error, 𝐸_𝑙²=ℎ²_𝑖𝑗𝜙^𝑁_𝑖𝑗−Φ^𝑁_𝑖𝑗^21/2, (53) where 𝐱_𝑖𝑗∈Ω_ℎ and 𝑡^𝑁=𝜋/4.

Figure 1 illustrates convergence of BDF1 and BDF2 using numerical characteristics and using CI (Figure 1(a)) and CSI (Figure 1(b)), respectively. The straight lines are the first-order line and the second-order line, respectively. We can see that BDF2 exhibits higher-order convergence than BDF1.

(a)

(b)

Figure 1 

𝐸𝑙2-error of BDF1 and BDF2 versus the time step Δ𝑡 in log-log scale, with 𝑇=𝜋/4, 𝜈=1.00×10−3; ℎ=Δ𝑡=0.1,0.05,0.04,0.25,0.02,0.005. CI are involved in (a) and CSI in (b).

Figure 2 shows the convergence of CN schemes using exact characteristics (Figure 2(a)) and using numerical characteristics (Figure 2(b)), respectively. We can see that when first-order numerical characteristics are involved, the computed result is poor. This confirms the fact that the higher-order numerical characteristic line is necessary to ensure reasonable convergence.

(a)

(b)

Figure 2 

𝐸𝑙2-error of CN scheme versus Δ𝑡 in log-log scale, with 𝑇=𝜋/4, 𝜈=1.25×10−4; ℎ=Δ𝑡=0.1,0.05,0.04,0.25,0.02. Exact characteristic lines are involved in (a) and numerical ones in (b).

Figure 3 exhibits convergence of BDF2 with the smaller diffusion coefficient 𝜈=1.25×10⁻⁴ using numerical characteristics. We see that either using CI or CSI, convergence is approximately second-order.

Figure 3 

𝐸𝑙2-errors of BDF2 using CI and CSI with 𝜈=1.25×10−4; 𝑇=𝜋/4, ℎ=Δ𝑡=0.1,0.08,0.05,0.02,0.008,0.005.

Remark 5. As the particular cases of the general formulation, the following schemes are also tested: 𝛼₂=2,𝛼₁=−3,𝛼₀𝛽=1;₂=3/2,𝛽₁=−1/2,𝛽₀𝛼=0,₂=3/4,𝛼₁=−1/2,𝛼₀𝛽=−1/4;₂=3/2,𝛽₁=−7/4,𝛽₀=5/4. (54) Both of them exhibit similar convergence properties as the BDF2 schemes.

6. Conclusions

In this paper we formulate a general SL scheme of second-order. By the MOL approach, this formulation includes all previously introduced schemes. We prove that this general scheme is second-order accurate even if the first-order characteristic line is involved. This formulation is very easy to extend to higher order cases. We see that MMOC can be considered a special version of SL method. Convergence properties of SL finite difference schemes are numerically tested.

Appendix

Transformation of Derivatives

In this appendix we deal with the differentiation of composite functions. When a convection-diffusion equation is written to the Lagrangian form, directional derivatives appears. However, for spatial discretizations we need transformation of the directional derivatives.

Let us give the transformation of derivatives for the diffusion term resulting from SL method.

Let 𝑋=𝑋(𝑥,𝑦), 𝑌=𝑌(𝑥,𝑦) and 𝑤(𝑋,𝑌)=𝑤(𝑋(𝑥,𝑦),𝑌(𝑥,𝑦)). By the chain rule, 𝜕𝑤(𝑋,𝑌)=𝜕𝑥𝜕𝑤⋅𝜕𝑋𝜕𝑋+𝜕𝑥𝜕𝑤⋅𝜕𝑌𝜕𝑌,()𝜕𝑥𝜕𝑤𝑋,𝑌=𝜕𝑦𝜕𝑤⋅𝜕𝑋𝜕𝑋+𝜕𝑦𝜕𝑤⋅𝜕𝑌𝜕𝑌.𝜕𝑦 (A.1) In the sequel, we simply write 𝑤(𝑋(𝑥,𝑦),𝑌(𝑥,𝑦)) as 𝑤. From (A.1) it follows that 𝜕𝑤=()⋅−⋅()𝜕𝑋𝜕𝑤/𝜕𝑥𝜕𝑌/𝜕𝑦𝜕𝑤/𝜕𝑦𝜕𝑌/𝜕𝑥(𝜕𝑋/𝜕𝑥)⋅𝜕𝑌/𝜕𝑦−𝜕𝑋/𝜕𝑦⋅(𝜕𝑌/𝜕𝑥),𝜕𝑤=()⋅−⋅()𝜕𝑌𝜕𝑤/𝜕𝑥𝜕𝑋/𝜕𝑦𝜕𝑤/𝜕𝑦𝜕𝑋/𝜕𝑥(𝜕𝑌/𝜕𝑥)⋅𝜕𝑋/𝜕𝑦−𝜕𝑌/𝜕𝑦⋅(𝜕𝑋/𝜕𝑥). (A.2)(A.3) By differentiating both sides of (A.2) with respect to 𝑥, and noting that 𝜕𝜕𝑥𝜕𝑤=𝜕𝜕𝑋²𝑤𝜕𝑋²⋅𝜕𝑋+𝜕𝜕𝑥²𝑤⋅𝜕𝑋𝜕𝑌𝜕𝑌,𝜕𝑥 (A.4) we have 𝜕²𝑤𝜕𝑋²⋅𝜕𝑋+𝜕𝜕𝑥²𝑤⋅𝜕𝑋𝜕𝑌𝜕𝑌=𝜕𝜕𝑥₂𝑤𝜕𝑥²⋅𝜕𝑌+𝜕𝑦𝜕𝑤⋅𝜕𝜕𝑥²𝑌−𝜕𝜕𝑥𝜕𝑦²𝑤⋅𝜕𝑥𝜕𝑦𝜕𝑌−𝜕𝑥𝜕𝑤⋅𝜕𝜕𝑦²𝑌𝜕𝑥²⋅𝜕𝑋⋅𝜕𝑥𝜕𝑌−𝜕𝑦𝜕𝑋⋅𝜕𝑦𝜕𝑌𝜕𝑥⁻¹+𝜕𝑤⋅𝜕𝑥𝜕𝑌−𝜕𝑦𝜕𝑤⋅𝜕𝑦𝜕𝑌⋅𝜕𝜕𝑥₂𝑋𝜕𝑥²⋅𝜕𝑌+𝜕𝑦𝜕𝑋⋅𝜕𝜕𝑥²𝑌−𝜕𝜕𝑥𝜕𝑦²𝑋⋅𝜕𝑥𝜕𝑦𝜕𝑌−𝜕𝑥𝜕𝑋⋅𝜕𝜕𝑦²𝑌𝜕𝑥²⋅𝜕𝑋⋅𝜕𝑥𝜕𝑌−𝜕𝑦𝜕𝑋⋅𝜕𝑦𝜕𝑌𝜕𝑥⁻². (A.5) Similarly, by differentiating both sides of (A.2) with respect to 𝑦, and noting that 𝜕𝜕𝑦𝜕𝑤=𝜕𝜕𝑋²𝑤𝜕𝑋²⋅𝜕𝑋+𝜕𝜕𝑦²𝑤⋅𝜕𝑋𝜕𝑌𝜕𝑌,𝜕𝑦 (A.6) we have 𝜕²𝑤𝜕𝑋²⋅𝜕𝑋+𝜕𝜕𝑦²𝑤⋅𝜕𝑋𝜕𝑌𝜕𝑌=𝜕𝜕𝑦₂𝑤⋅𝜕𝑥𝜕𝑦𝜕𝑌+𝜕𝑦𝜕𝑤⋅𝜕𝜕𝑥²𝑌𝜕𝑦²−𝜕²𝑤𝜕𝑦²⋅𝜕𝑌−𝜕𝑥𝜕𝑤⋅𝜕𝜕𝑦²𝑌⋅𝜕𝑥𝜕𝑦𝜕𝑋⋅𝜕𝑥𝜕𝑌−𝜕𝑦𝜕𝑋⋅𝜕𝑦𝜕𝑌𝜕𝑥⁻¹+𝜕𝑤⋅𝜕𝑥𝜕𝑌−𝜕𝑦𝜕𝑤⋅𝜕𝑦𝜕𝑌⋅𝜕𝜕𝑥₂𝑋⋅𝜕𝑥𝜕𝑦𝜕𝑌+𝜕𝑦𝜕𝑋⋅𝜕𝜕𝑥²𝑌𝜕𝑦²−𝜕²𝑋𝜕𝑦²⋅𝜕𝑌−𝜕𝑥𝜕𝑋⋅𝜕𝜕𝑦²𝑌⋅𝜕𝑥𝜕𝑦𝜕𝑋⋅𝜕𝑥𝜕𝑌−𝜕𝑦𝜕𝑋⋅𝜕𝑦𝜕𝑌𝜕𝑥⁻². (A.7) Eliminating the mixed partial derivatives from the left-hand sides of both (A.5) and (A.7), we have 𝜕²𝑤𝜕𝑋²⋅𝜕𝑋⋅𝜕𝑥𝜕𝑌−𝜕𝑦𝜕𝑋⋅𝜕𝑦𝜕𝑌=𝜕𝜕𝑥₂𝑤𝜕𝑥²⋅𝜕𝑌𝜕𝑦²+𝜕𝑤²𝜕𝑦²⋅𝜕𝑌𝜕𝑥²−2𝜕²𝑤⋅𝜕𝑥𝜕𝑦𝜕𝑌⋅𝜕𝑥𝜕𝑌⋅𝜕𝑦+⋯𝜕𝑋⋅𝜕𝑥𝜕𝑌−𝜕𝑦𝜕𝑋⋅𝜕𝑦𝜕𝑌𝜕𝑥⁻¹+⋯, (A.8) where the omitted terms consist of the second derivatives of 𝑋 or 𝑌 with respect to 𝑥, 𝑦. If 𝑋 and 𝑌 are polynomials of 𝑥,𝑦 of degrees not more than one (as the example in Section 5), these terms will become zeros.

Similarly, from (A.3), we have 𝜕²𝑤𝜕𝑌²⋅𝜕𝑌⋅𝜕𝑥𝜕𝑋−𝜕𝑦𝜕𝑌⋅𝜕𝑦𝜕𝑋=𝜕𝜕𝑥₂𝑤𝜕𝑥²⋅𝜕𝑋𝜕𝑦²+𝜕𝑤²𝜕𝑦²⋅𝜕𝑋𝜕𝑥²−2𝜕²𝑤⋅𝜕𝑥𝜕𝑦𝜕𝑋⋅𝜕𝑥𝜕𝑋⋅𝜕𝑦+⋯𝜕𝑌⋅𝜕𝑥𝜕𝑋−𝜕𝑦𝜕𝑌⋅𝜕𝑦𝜕𝑋𝜕𝑥⁻¹+⋯. (A.9) Manipulation of (A.8) and (A.9) leads to 𝜕²𝑤𝜕𝑋²+𝜕²𝑤𝜕𝑌²=𝜕²𝑤𝜕𝑥²⋅𝜕𝑋𝜕𝑦²+𝜕𝑌𝜕𝑦²+𝜕𝑤²𝜕𝑦²⋅𝜕𝑌𝜕𝑥²+𝜕𝑋𝜕𝑥²−2𝜕²𝑤⋅𝜕𝑥𝜕𝑦𝜕𝑌⋅𝜕𝑥𝜕𝑌+𝜕𝑦𝜕𝑋⋅𝜕𝑥𝜕𝑋⋅𝜕𝑦+⋯𝜕𝑌⋅𝜕𝑥𝜕𝑋−𝜕𝑦𝜕𝑌⋅𝜕𝑦𝜕𝑋𝜕𝑥⁻²+⋯, (A.10) where the omitted terms consist of the second derivatives of 𝑋 or 𝑌.

If 𝑋(𝑥,𝑦),𝑌(𝑥,𝑦) are polynomials of 𝑥,𝑦 of degrees not more than one, then from (A.10) it follows that 𝜕²𝑤𝜕𝑋²+𝜕²𝑤𝜕𝑌²=𝜕²𝑤𝜕𝑥²⋅𝜕𝑋𝜕𝑦²+𝜕𝑌𝜕𝑦²+𝜕𝑤²𝜕𝑦²⋅𝜕𝑌𝜕𝑥²+𝜕𝑋𝜕𝑥²−2𝜕²𝑤⋅𝜕𝑥𝜕𝑦𝜕𝑌⋅𝜕𝑥𝜕𝑌+𝜕𝑦𝜕𝑋⋅𝜕𝑥𝜕𝑋⋅𝜕𝑦𝜕𝑌⋅𝜕𝑥𝜕𝑋−𝜕𝑦𝜕𝑌⋅𝜕𝑦𝜕𝑋𝜕𝑥⁻². (A.11) Furthermore, if we use the exact characteristic line 𝑋𝑌𝑥,𝑦,𝑡=𝑥cos𝑡−𝑦sin𝑡,𝑥,𝑦,𝑡=𝑥sin𝑡+𝑦cos𝑡, (A.12) then from (A.11), it holds that 𝜕²𝑤𝜕𝑋²+𝜕²𝑤𝜕𝑌²=𝜕²𝑤𝜕𝑥²+𝜕𝑤²𝜕𝑦². (A.13) If we use the numerical characteristic line 𝑋𝑌𝑥,𝑦=𝑥+𝑦Δ𝑡,𝑥,𝑦=𝑦−𝑥Δ𝑡, (A.14) then from (A.11), it holds that 𝜕²𝑤𝜕𝑋²+𝜕²𝑤𝜕𝑌²=𝜕²𝑤/𝜕𝑥²+𝜕𝑤²/𝜕𝑦²1+Δ𝑡². (A.15)

Acknowledgments

This work is supported by the Major State Research Program of China Grant no. 1999032803, the Natural Science Foundation of Shandong Grants no. ZR2010AL021 and ZR2010AQ010, and the Project of Shandong Province Higher Educational Science and Technology Program Grant no. J11LA09. The authors would like to express their gratitude to Professor Yirang Yuan for the help and the reviewers for the valuable comments and opinions.