Convergence, Consistency, and StabilityDefinitionA one-step finite difference scheme approximating a partial differential equationis a convergent scheme if for any solution to the partial differential equation,u(t, x), and solutions to the finite difference scheme, vni, such that v0iconvergesto u0(x) as i∆x converges to x, then vniconverges to u(t, x) as (n∆t, i∆x) con-verges to (t, x) as ∆t, ∆x converge to 0.DefinitionGiven a partial differential equation P u = f and a finite difference scheme,P∆t,∆xv = f , we say that the finite difference scheme is consistent with thepartial differential equation if for any smooth function φ(x, t)P φ − P∆t,∆xφ → 0 as ∆t, ∆x → 0.ExampleConsider the one-way wave equation given by the operator P = ∂/∂t + α∂/∂xP φ = φt+ αφxwith α greater than 0. We will evaluate the consistency of the forward-timeforward-space scheme with difference operator P∆t,∆xgiven byP∆t,∆xφ =φn+1i− φni∆t+ αφni+1− φni∆x(1)whereφni= φ(n∆t, i∆x).We begin by taking the Taylor expansion of the function φ in t and x about(tn, xi). We have thatφn+1i= φni+ ∆tφt+12∆t2φtt+ O(∆t3)φni+1= φni+ ∆xφx+12∆x2φxx+ O(∆x3).This gives usP∆t,∆xφ = φt+ αφx+12∆tφtt+ α12∆xφxx+ O(∆t2) + O(∆x2)ThusP φ − P∆t,∆xφ =12∆tφtt+12∆xφxx+ O(∆t2) + O(∆x2)→ 0 as (∆t, ∆x) → 0.Thus, this scheme is consistent. 1DefinitionThe L2-norm of a grid function w, denoted by kwk∆x, is defined askwk∆x= ∆x∞Xm=−∞|wm|2!1/2.DefinitionA finite difference scheme P∆t,∆xvni= 0 for a first-order equation is stable ina stability region Λ if there is an integer J such that for any positive time T ,there is a constant CTsuch thatkvnk∆x≤ CTJXj=0kvjk∆xfor 0 ≤ n∆t ≤ T , with (∆t, ∆x) ∈ Λ.Von Neumann AnalysisProving stability directly from the definition is quite difficult, in general. In-stead, it is easier to use tools from Fourier analysis to evaluate the stability offinite difference schemes. In particular, it can be shown that, for some solutionto a finite difference scheme vn, there is a simple mathematical relationshipbetween the Fourier transforms ˆvn(ξ) and ˆv0(ξ) given byˆvn(ξ) = g(∆xξ, ∆t, ∆x)nˆv0(ξ)where g(∆xξ, ∆t, ∆x) = g(θ, ∆t, ∆x) is called the amplification factor and ˆvn(ξ)is the amplitude of the frequency ξ in the solution vn(note that the second su-perscript n in the above equation is a power and not an index). This quantityis so named because it represents the amount that each frequency in the solu-tion is am plified in advancing the solution one time step. That this relationshiparises from such analysis should not be surprising: the fundamental power ofFourier transforms is that they transform differentiation in the temporal domainto multiplication in the frequency domain.DefinitionA one-s tep finite difference scheme with constant coefficients is stable in a sta-bility region Λ if and only if there is a constant K (independent of θ , ∆t, and∆x) such that|g(θ , ∆t, ∆x)| ≤ 1 + K∆twith (∆t, ∆x) ∈ Λ. If g(θ, ∆t, ∆x) is independent of ∆t and ∆x, the stabilitycondition may be replaced with the restricted stability condition|g(θ )| ≤ 1.2ExampleConsider again the one-way wave equation and let us evaluate the stability of theforward-time forward-space scheme given in equation (1). It can be shown thatall solutions of any one-step difference scheme will have the form vnj= gneijθ. Toanalyze the stability of such a scheme, we may therefore substitute accordinglyand solve for g.gn+1eijθ− gneijθ∆t+ αgnei(j+1)θ− gneijθ∆x= 0gneijθ(g − 1)∆t= −αgneijθ(eiθ− 1)∆x(g − 1) = −α(eiθ− 1)∆t∆xg = 1 + α∆t∆x− α∆t∆xeiθg = 1 + αλ − αλeiθwhere λ = ∆t/∆x and α is positive. We can then calculate|g|2= 1 + 4αλ(1 + αλ) sin212θ.Since λ is constant, we may use the restricted stability condition and we seethat |g| is greater than 1 for θ 6= 0. Therefore, this scheme is unstable. The Lax-Richtmyer Equivalence TheoremThe Lax-Richtmyer Equivalence Theorem is often called the Fundamental Theo-rem of Numerical Analysis, even though it is only applicable to the small subsetof linear numerical methods for well-posed, linear partial differential equations.Along with Dahlquist’s equivalence theorem for ordinary differential equations,the notion that the relationshipconsistency + stability ⇐⇒ convergencealways holds has c aused a great deal of confusion in the numerical analysisof differential equations. In the case of PDEs, mathematicians are most of-ten interested in nonlinear phenomena, for which Lax-Richtmyer does not ap-ply. More damningly, the forward implication that consistency + stability =⇒convergence is trivial for linear schemes, and thus it is only the converse notionthat convergence =⇒ stability that the theorem contributes. The intuitionthat the theorem gives for problems that fall outside the scope of Lax-Richtmyer,however, is faulty, since consistency and stability are often insufficient for con-vergence, and convergence need not imply stability in general.3Supp ose we have a stable, linear numerical method that is consistent with awell-posed linear partial differential equation. Suppose further that the globalerror at time T = N∆t is the grid function denoted byEN= QN− qNwhere Qnis the numerical approximation to the solution and qnis the exactsolution ∀n ≤ N. We will denote our method by N (·), such thatQn+1= N (Qn).We claim that it is straightforward to show that the method is convergent,which amounts to proving that kENk → 0 as ∆t → 0 and N∆t → T .Since N is stable, we know that there exists a constant CTdepending onlyon T such thatkNnk ≤ CT, ∀n ≤ N = T /∆t.We can also express the local trunctation error as the one step error divided bythe time step, orτn=1∆tN (qn) − qn+1 ,and note that consistency implies that τn→ 0 as ∆t → 0.We can derive a simple recurrence for the error at time n + 1:En+1= Qn+1− qn+1= N (Qn) − qn+1= N (En+ qn) − qn+1= N (En+ qn) − N (qn) + N (qn) − qn+1= N (En+ qn) − N (qn) + 4t τn= N (En) + 4t τnwhere we use the linearity of N (·) in the last equality. From this formulation,it is simple to examine the norm of the error and argue that the requirementsfor convergence are met by stability and