This finishes the presentation of the scheme. However, we have one outstanding item (i) is the schemestable?5.1.2von Neumann Stability Analysisvon Neumann was a pioneer of mathematics, physics and computing. He developed a method of stabilityanalysis which may be applied easily to the discretised form of the governing equation/s. This requires thefollowing steps:1. Assume a solution to the governing equation which can represent a range of the possible physicalbehaviour2. Substitute this into the discretised governing equations, and simplify
f(x)=∞∑k=−∞ckeIknx(62)where it should be noted that from Euler’s Formula:eIknx=cos(knx)+Isin(knx),e−Iknx=cos(knx)−Isin(knx)(63)whereckis a complex wave amplitude andk=πnLis the wave number.By lettingnbe the temporalindex with incrementΔtandjbe the spatial index with incrementΔx, the discrete version of the complexexponential Fourier Series can be written as:u(x)=∞∑k=−∞ckeIkniΔx(64)Using this, the solution to the PDE is now written as:u(x,t)=∞∑k=−∞GneIkniΔx(65)whereGn(k,t)is the time dependent amplitude of the mode. This will capture the potential oscillatory,exponentially decaying, or exponentially growing behaviour of the PDE solution in time. If|G|>1 thenthe amplitude will grow exponentially in time, if|G|=1 the amplitude will not grow, and if|G| ≤1 theamplitude will decrease. Thus a stable scheme has|G| ≤1, and the boundary of stability is|G|=1.For linear PDEs in this course, all modes demonstrate the same physical behaviour. This means that byunderstanding the behaviour of one single mode, you can understand the behaviour of all the modes in thesolution. Thus in von Neumann analysis we assume that the solution consists of just one single mode, thus:uni=GneIkniΔx(66)Step 2: Substitution into the Discretised EquationStarting from Eqn. (61), substitute in Eqn. (66), giving:Gn+1eIkniΔx=GneIkniΔx+c2ΔtΔx2parenleftBigGneIkni+1Δx−2GneIkniΔx+GneIkni−1ΔxparenrightBig(67)Substantial simplification can be made by dividing through both sides byGneIkniΔx.