16.920J/SMA 5212 Numerical Methods for PDEs There is a total of N+I grid points such that x=jAx, j=0,1,2 Side 4 STABILITY ANALYSIS Discretization Use the Central Difference scheme for a-u which is second-order accurate Schemes of other orders of accuracy may be constructed Construction of spatial Difference Scheme of Any Order p The idea of constructing a spatial difference operator is to represent the spatial differential operator at a location by the neighboring nodal points, each with its own The order of accuracy, p of a spatial difference scheme is represented as O(ArP) Generally, to represent the spatial operator to a higher order of accuracy, more nodal j2-1 户+1j+2 Consider the following procedure of determining the spatial operator dx cyo(△x2)16.920J/SMA 5212 Numerical Methods for PDEs 3 Slide 4 STABILITY ANALYSIS Discretization which is second-order accurate. • Schemes of other orders of accuracy may be constructed. Slide 5 Construction of Spatial Difference Scheme of Any Order p The idea of constructing a spatial difference operator is to represent the spatial differential operator at a location by the neighboring nodal points, each with its own weightage. The order of accuracy, p of a spatial difference scheme is represented as ( ) p O ∆x . Generally, to represent the spatial operator to a higher order of accuracy, more nodal points must be used. Consider the following procedure of determining the spatial operator j du dx ￾ ✁ ✂ ✄ ☎ ✆ up to the order of accuracy ( ) 2 O ∆x : There is a total of 1 grid points such that , 0,1,2,...., N j x j x j N + = ∆ = 2 U 2 se the Central Difference Scheme for u x ∂ ∂ 2 1 1 2 2 2 2 ( ) j j j j u u u u O x x x + − + − ✝ ✞ ∂ = + ∆ ✟ ✠ ∂ ∆ ✡ ☛ j−2 • j−1 • j • j+1 • j+2 • j du dx ☞ ✌ ✍ ✎ ✏ ✑