Following the derivation in we consider the stochastic differential equation 1∂u∂t(x,t)=D[u]+Ps[u], where u=u(x,t) is a function of spatial and temporal variables. The term D[u] contains only deterministic terms and all of the stochastic terms are in Ps[u]. R˙(t) is a colored noise term which is spatially homogeneous. Without loss of generality, we assume E[R˙(t)]=0. If Ps[u] has the form 2Ps[u]=g[u]R˙(t), the GIC at the jth time step in differential form is given by Ij with 3Ij=12∂(g[u])∂ug[u]|tjE[ΔRj2], where tj is the time at the jth time step, ΔRj is the jth increment of R and ∂(g[u])∂ug[u] is the Fréchet derivative operator applied to g[u]. The GIC is applied to a numerical integration scheme converging to the Itô solution as follows. At each time step of the numerical integration, Ij is computed and added to the numerical solution of the right-hand-side of Eq. () for that timestep. The only term in Ij that needs to be computed at each time step is ∂(g[u])∂ug[u]. We note that in the white noise case, where E[ΔRj2]=Δt, the GIC is equal to the Itô correction (see for more details).

We use the following approximation n(t) of the noise process R˙(t) from . 4n(t)=1NfΔtC(ω0)b02+∑m=1NfC(ωm)[amsin⁡(ωmt)+bmcos⁡(ωmt)],C(ω)=e-αω2,ωm=2πm(N-1)Δt,Nf=(N-1)/2, where N is the number of discrete time levels per unit of time, including the starting and ending time levels. The parameter α controls the color of the Fourier spectrum of n(t) (α=0 corresponds to white noise while α≠0 to colored noise). To construct different realizations of the noise process, we sample, for m=0,…,Nf, the coefficients am and bm independently, from the normal distribution N(0,1) for different initial seeds of a random number generator. Different resolutions of the same noise path can be computed either by computing the noise for the smallest Δt and then sub sampling, or by computing the noise for each choice of Δt separately with the same initialization of the random coefficients am and bm. It should be noted that n(t) is an approximate random noise. The true noise term R˙(t) would contain an infinite number of Fourier modes while n(t) only has a finite number of modes. The approximation n(t) is therefore smooth in t but not resolved by a sampling of N discrete time points. In the white noise case, when α=0, the coefficient C(wm) is constant for all frequencies wm, approximating the constant power spectral density of white noise. In the case of colored noise, when α≠0, the coefficient C(wm) dampens higher frequency terms resulting in a colored noise spectrum. As Nf→∞ the approximation n(t) converges in L2 to the white noise process R˙ when α=0 according to the Kosambi–Karhunen–Loève theorem . When α≠0 n(t) converges to a smooth function as higher frequency oscillations are damped by the coefficients C(wm). These properties make n(t) a suitable approximation of R˙(t) in numerical modeling. It should also be noted that the approximation n(t) is constant in space. This choice was made for convenience and to directly follow the work of that inspired this study. The GIC is equally valid for spatially varying noise processes.

For the first two cases investigated in this work, the coefficient function g[u] in Eq. () will take the form g[u]=∂u∂x. For this choice of g[u], we have 5∂(g[u])∂ug[u]=∂2u∂x2, and for this noise approximation n(t) we have 6E[ΔRj2]≈E[n(tj)2]=lim⁡Nf→∞1NfC(ω0)22+∑m=1NfC(ωm)2. With the true noise process n(t) represented by an infinite number of Fourier modes, the GIC in differential form is given by 7Ij=12∂2u∂x23lim⁡Nf→∞1NfC(ω0)22+∑m=1NfC(ωm)2. We note that Ij→0 as N→∞ for any colored noise (α≠0), but not for the white noise (α=0). This is again because any colored noise can be resolved by taking sufficiently small time steps, at which point the system can be understood as a deterministic one, and the GIC is no longer necessary. As we can only use a finite number of modes numerically as in Eq. (), the GIC in differential form we use in this work is 8Ij=12∂2u∂x2tj1NfC(ω0)22+∑m=1NfC(ωm)2.

Following Eq. (2.7) and the boundary conditions specified in , we study the following modified Korteweg-de Vries (KdV) equation for the amplitude A=A(x,t) of low frequency atmospheric waves: 9∂A∂t=-md∂3A∂x3+(mp(x)+mnA)∂A∂x+mg(x)A. The above equation is modified from the traditional KdV equation with the addition of the linear growth term mg(x)A. The dispersion and nonlinear coefficients, md and mn, are constant, whereas the linear, long-wave phase speed and growth/decay coefficients, mp(x) and mg(x), are functions of the zonally varying background flow.

We introduce the stochastic perturbation n(t) on the linear advection term of Eq. () as 10∂A∂t=-md∂3A∂x3+(mp(x)+mnA+n(t))∂A∂x+mg(x)A. Here the coefficient function g in Eq. () is g[A]=∂A/∂x and the GIC correction at each time step tj is given by 11IjA=12∂2A∂x2tj1NfC(ω0)22+∑m=1NfC(ωm)2.

Following , we study a nonrotating, stably stratified, nonhydrostatic Boussinesq fluid bounded above and below (in z direction) by rigid, horizontal boundaries, but periodic in the horizontal (x) direction. The governing equations along an x-z cross-section may be combined into two equations in two unknowns, the vorticity ζ and the potential temperature θ, as 12∂ζ∂t=-u∂ζ∂x+w∂ζ∂z+g0θ0∂θ∂x+F,13∂θ∂t=-u∂θ∂x+w∂θ∂z+w∂θ0∂z+H, along with the relations 14u=∂ψ∂z,w=-∂ψ∂x,ζ=∇2ψ, where ∇2 is the Laplacian operator in the x-z plane, and u, w, ψ, F, H, g0, and θ0 are the zonal wind, vertical wind, geostrophic pressure (stream function) field, vorticity source, heat source, standard acceleration due to gravity, and reference temperature, respectively (see Appendix D of for more details about the sub-grid parameterizations F and H, linear advection, the eddy viscosity, the thermal diffusion, initial conditions, and boundary conditions etc.).

For this study we do not include stochastic perturbations of F or H as in and introduce a stochastic perturbation n(t) on the zonal wind u in Eqs. () and () as, 15∂ζ∂t=-(u+n)∂ζ∂x+w∂ζ∂z+gθ0∂θ∂x+F,16∂θ∂t=-(u+n)∂θ∂x+w∂θ∂z+w∂θ0∂z+H. Here, the function g in Eq. () is g[ζ]=∂ζ/∂x in Eq. () and g[θ]=∂θ/∂x in Eq. (), and the GICs are given by 17Ijζ=12∂2ζ∂x2tj1NfC(ω0)22+∑m=1NfC(ωm)2,18Ijθ=12∂2θ∂x2tj1NfC(ω0)22+∑m=1NfC(ωm)2 respectively. We make one additional change to the model: where used a Δt dependent hyper-diffusion parameter we set this parameter to be constant; this is necessary for convergence analysis.

To demonstrate the effect of the GIC on higher order schemes, we consider, for simplicity, a homogeneous differential equation driven purely by stochastic terms (drift-free) given by 19dX(t)dt=X(t)n(t), with initial condition X(0)=X0. For this choice of g and the noise process n(t), the GIC is given by 20Ij=12X|tj1NfC(ω0)22+∑m=1NfC(ωm)2 Numerical results are presented later in the paper for the three time integration schemes summarized below.

The order 1.0 Milstein (MS) scheme for Eq. () with colored noise can be written as 21Yn+1=Yn+Ynn(tn)Δt+12Yn(n(tn)Δt)2-E[(ΔRj)2)], with the approximation Yn≈X(tn). In the case of white noise, where E[(ΔRj)2)]=Δt, Eq. () reduces to the traditional MS scheme. This scheme was chosen to demonstrate the effects of adding GIC to a scheme which converges to the Itô solution with order 1.0 where the forward Euler scheme previously used converges with order 0.5.

Schemes for strong and weak approximation with colored noise are discussed in detail in . It will sometimes be the case that the order of convergence of a scheme is changed by the addition of the GIC. To demonstrate this, we consider the order 1.5 Strong Taylor Scheme detailed in section 10.4.1; for brevity we shall call this the KP scheme. We will demonstrate that while this scheme converges to the Itô solution with order 1.5 the addition of the GIC results in a scheme converging to the Stratonovich solution with order 1. The KP scheme for Eq. () with colored noise can be written as 22Yn+1=Yn+Ynn(tn)Δt+12Yn(n(tn)Δt)2-E[(ΔRj)2)]+12Yn13(n(tn)Δt)2-E[(ΔRj)2)]n(tn)Δt. We note that for both of these schemes, there is a term equivalent to the GIC with a minus sign. It is therefore more efficient to remove this term and the GIC, rather than redundantly subtracting and adding it.

The third higher order scheme we consider is the Milstein scheme with the highest order derivative approximated with a forward difference as described in Sect. 11.1.3, we will call this scheme KP2. This scheme was chosen to demonstrate how the GIC cam be applied to multi-step methods. The KP2 scheme for Eq. () with colored noise can be written as 23Yn+1=Yn+Ynn(tn)Δt+12ΔtỸn-Yn(n(tn)Δt)2-E[(ΔRj)2)] with supporting value 24Ỹn=Yn+YnΔt. When adding the GIC to KP2 we add it only to Eq. (). Unlike the previous two schemes, there is no term equivalent to the subtraction of the GIC, so the GIC must be added explicitly to Eq. (). A summary of all numerical schemes considered in this study and their relevant properties is provided in Table .

In this experiment, we compare the convergence and final error for forward Euler with and without the GIC for varying colors of the noise. For the 1D KdV model, we use RK2 with Δtr=10-5 for the reference solution. In Fig. , we integrate for 5 units of time and compare results over 100 realizations of the noise for each of the chosen colors α=10-3,10-4,10-5,10-6,10-7,10-8,10-9,10-10, 10-11,10-12]. Table  contains a summary of the parameters used in this experiment. As expected, for α=0, forward Euler never begins converging as it converges to the Itô interpretation instead of that of Stratonovich. The addition of the generalized Itô correction, in this case, equals the classical Itô correction and results in convergence of order 0.5 in accordance with the theory of . For all non-zero α, we see that for sufficiently small Δt forward Euler will begin to converge with order 1.0 as the noise is sufficiently resolved in time that the system can be understood as purely deterministic. For non-zero α, the addition of the GIC reduces the final error and increases the critical time step after which the scheme achieves the theoretic best convergence rate of 1.0. The critical time step is increased with the addition of the GIC, which acts as a dissipative term at each point in space, functioning similar to smoothing the noise in time. For time steps which do not resolve the colored noise, the GIC smooths the solution which better represents the underlying smoothness of the colored noise.

For the 2D fluid model Eqs. () and (), we use RK2 with a time step Δtr=10-5 as the reference solution. We evolve the model deterministically until t=1000 with the initial conditions prescribed in to guarantee that the fluid is sufficiently evolved from the initial condition. In Fig. , we compare results for forward Euler with and without the GIC over 100 different realizations of the noise for different colors α=[0,10-8,10-6,10-4,10-2] integrated to t=1001 with time steps ranging from 0.5 to 10-5. Table  contains a summary of the parameters used in this experiment. Again α=0 demonstrates the well-known classical Itô correction for white noise and for non-zero α addition of the GIC results in a lower final error and increased maximal time steps for which the scheme achieves the maximal convergence of order 1.0.

In this experiment, we scale the magnitude (equivalently the variance) of the noise with a multiplicative factor γ creating a new noise term 26g*[u]=γg[u]n(t). Figure shows results averaged over 100 realizations of the colored noise term for the 2D fluid model with α=10-10, γ=[0.01,0.1,0.2,0.5,1,2,5,10] integrated over 1 unit of time with the same initial condition as in Sect. . We see that for all choices of γ, the critical Δt for convergence remains the same Δtc=10-4. Although the critical Δt is unchanged, we see that the relative error increases with increasing γ. This is exactly as expected as the leading error term between RK2 and forward Euler with the GIC is O(γ2). For brevity, we omit this proof. Although the GIC improves the convergence of forward Euler over the range of magnitudes demonstrated in Fig. , there is of course a maximal γ such that for any larger γ the scheme becomes unstable.

As forward Euler is a very simple scheme and not a commonly used we next demonstrate how the GIC may be applied to higher order methods. We use the analytic Stratonovich solution to Eq. () for computing errors in this section in place of the reference solutions used in previous sections.

Figure shows the convergence rate of scheme Eq. () to the Stratonovich solution with and without the GIC for different colors of noise. Similar to the previous results for forward Euler, the Millstein scheme converges to the Stratonovich solution for all colors of noise and does not converge in the case of white noise, where it converges to the Itô solution. The addition of the GIC decreases the final error for all colors of noise and changes the convergence in the white noise case to the Stratonovich solution. In this case, we see that with the addition of the GIC, the convergence in the white noise case remains order 1.0.

Figure shows the convergence rate of the KP scheme Eq. () to the Stratonovich solution with and without the GIC for different colors of noise. Again we see that the addition of the GIC reduces the final error and improves the convergence rate for colored noise and changes the convergence of the white noise case to the Stratonovich solution. Note that in the white noise case, the order 1.5 convergence of the KP scheme to the Itô solution has been changed to order 1.0 convergence to the Stratonovich solution. The GIC is an order Δt correction term and does not “correct” the higher order terms. Higher order corrections would be necessary to construct an order 1.5 scheme in this manner. A summary of all numerical schemes considered in this study and their orders of convergence is provided in Table .

Figure shows the convergence rate of the KP2 scheme Eq. () to the Stratonovich solution with and without the GIC for different colors of noise. Again we see that the addition of the GIC reduces the final error and improves the convergence rate for colored noise and changes the convergence of the white noise case to the Stratonovich solution. We note again that although two separate calculations are performed, one for the supporting value Eq. () and one for the next time step Eq. (), the GIC is only added to the computation of Eq. ().

In this final experiment, we demonstrate the computational efficiency of the GIC. We average the run time of all three schemes considered on the 2D model Eqs. ()–() over 100 realizations of white noise (γ=1) integrated over 50 units of time with Δt=[1,10-1,10-2,10-3]. Figure  shows the ratio of the run time of RK2 / Euler and Euler-GIC / Euler. As forward Euler is a first-order scheme (in time) and RK2 is a second-order scheme the ratio of RK2 / Euler should be around 2 once enough time steps are taken, this is shown in the orange curve. The ratio of Euler+GIC / Euler remains below 1.1 (blue curve), demonstrating that Euler+GIC remains similar in computational cost. We note that we are solving this model with a pseudo-spectral technique, therefore the additional computation for the GIC at each time step consists of two additional multiplications to compute ∂2ζ∂x2 and ∂2θ∂x2 in frequency space. In general, the efficiency of computing the GIC for a given model is determined by the difficulty of computing (or approximating) ∂g[u]∂ug[u] at each time step compared with the computation of the right hand side of Eq. ().