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==== Stability and dynamics ====
[[File:Figure ODE.jpg|thumb|488x488px|'''Fig. 1''' Phase portrait (dashed line) and approximate phase portrait (solid line) of the nonlinear ODE (4.10)-(4.11) computed by the order 2 LL scheme (4.2), the order 4 classical Rugen-Kutta scheme [[Runge–Kutta methods|''RK''4]], ''and the order 4 LLRK''4 schemes (4.8) with step size h=1/2 , and p=q=6.]] By construction, the LL and HOLL discretizations inherit the stability and dynamics of the linear ODEs, but it is not the case of the LL schemes in general. With <math>p\leq q\leq p+2</math>, the LL schemes (4.6)-(4.9) are [[Stiff equation|''A''-stable]].<ref name=":3" />. With ''q = p + 1 or q = p + 2'', the LL schemes (4.6)-(4.9) are also [[L-stability|''L''-stable]].<ref name=":3" />. For linear ODEs, the LL schemes (4.6)-(4.9) converge with order ''p + q'' <ref name=":3" /> <ref name= ″:25″ />. In addition, with ''p = q = 6'' and <math>m_{n}</math> ''= d'', all the above described LL schemes yield to the ″exact computation″ (up to the precision of the [[floating-point arithmetic]]) of linear ODEs on the current personal computers <ref name=":3" /> <ref name= ″:25″ /> .<ref name=":2" /> <ref name=":9" />. This includes [[Stiff equation|stiff]] and highly oscillatory linear equations. Moreover, the LL schemes (4.6)-(4.9) are regular for linear ODEs and inherit the [[Symplectic geometry|symplectic structure]] of [[Hamiltonian mechanics|Hamiltonian]] [[harmonic oscillator]]s <ref name= ″:25″ /> .<ref name=":2" /> <ref name=":9" />. These LL schemes are also linearization preserving, and display a better reproduction of the [[Stable manifold|stable and unstable manifolds]] around [[hyperbolic equilibrium point]]s and [[Limit cycle|periodic orbits]] that [[Numerical methods for ordinary differential equations|other numerical schemes]] with the same stepsize <ref name= ″:25″ /> .<ref name=":2" /> <ref name=":9" />. For instance, Figure 1 shows the [[phase portrait]] of the ODEs
<math>\frac{dx_{1}}{dt} =-2x_{1}+x_{2}+1-\mu f\left( x_{1},\lambda \right)
\mathbf{0}_{1\times (d+1)} & 1
\end{array}
\right]</math>, and ''p+q>1''. For large systems of RDEs ,<ref name=":10" />,
<div style="text-align: center;">
==== Stability and dynamics ====
By construction, the strong LL and HOLL discretizations inherit the stability and [[Random dynamical system|dynamics]] of the linear SDEs, but it is not the case of the strong LL schemes in general. LL schemes (7.2)-(7.5) with <math>p\leq q\leq p+2 </math> are ''A''-stable, including stiff and highly oscillatory linear equations.<ref name=":12" />. Moreover, for linear SDEs with [[Pullback attractor|random attractors]], these schemes also have a random attractor that [[Convergence in probability|converges in probability]] to the exact one as the stepsize decreases and preserve the [[ergodicity]] of these equations for any stepsize.<ref name=":4" /><ref name=":12" />. These schemes also reproduce essential dynamical properties of simple and coupled harmonic oscillators such as the linear growth of energy along the paths, the oscillatory behavior around 0, the symplectic structure of Hamiltonian oscillators, and the mean of the paths.<ref name=":4" /><ref name=":5">de la Cruz H.; Jimenez J.C.; Zubelli J.P. (2017). "Locally Linearized methods for the simulation of stochastic oscillators driven by random forces". BIT Numer. Math. 57: 123–151. [http://doi.org:10.1007%2Fs10543-016-0620-2 doi:10.1007/s10543-016-0620-2]. S2CID 124662762.</ref>. For nonlinear SDEs with small noise (i.e., (7.1) with <math>\mathbf{g}_{i}(t)\approx 0 </math>), the paths of these SLL schemes are basically the nonrandom paths of the LL scheme (4.6) for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the paths of the SLL scheme.<ref name=":4" />. For instance, Fig 4 shows the evolution of domains in the phase plane and the energy of the stochastic oscillator
<math>\begin{array}{ll}
==== Stability and dynamics====
[[File:Figure WSDE.png|thumb|429x429px|'''''Fig. 5''''' Approximate mean of the SDE (8.2) computed via Monte Carlo with ''100'' simulations of various schemes with ''h=1/16'' and ''p=q=6''.]] By construction, the weak LL discretizations inherit the stability and [[Random dynamical system|dynamics]] of the linear SDEs, but it is not the case of the weak LL schemes in general. WLL schemes, with <math>p\leq q\leq p+2,</math> preserve the [[Moment (mathematics)|first two moments]] of the linear SDEs, and inherits the mean-square stability or instability that such solution may have.<ref name=":11" />. This includes, for instance, the equations of coupled harmonic oscillators driven by random force, and large systems of stiff linear SDEs that result from the method of lines for linear stochastic partial differential equations. Moreover, these WLL schemes preserve the [[ergodicity]] of the linear equations, and are geometrically ergodic for some classes of nonlinear SDEs.<ref name=":6"> Hansen N.R. (2003) Geometric ergodicity of discre-time approximations to multivariate diffusion. Bernoulli.9 : 725-743 [[doi:10.3150/bj/1066223276]]</ref>. For nonlinear SDEs with small noise (i.e., (8.1) with <math>\mathbf{g}_{i}(t)\approx 0</math>), the solutions of these WLL schemes are basically the nonrandom paths of the LL scheme (4.6) for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the mean of the WLL scheme.<ref name=":11" />. For instance, Fig. 5 shows the approximate mean of the SDE
<math>dx=-t^{2}x\text{ }dt+\frac{3}{2(t+1)}e^{-t^{3}/3}\text{ }dw_{t},\qquad \qquad x(0)=1, \qquad \quad(8.2)</math>
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