High-Order, Finite-Volume Methods in Mapped Coordinates

where y is the vector of N unknowns (uJ) iand A(x) is an N × N spatiallyvaryingvariable-coefficient matrix. For all four-stage, fourth-order Runge-Kutta temporal discretizations, the characteristic polynomial isP (z j ) = 1 + z j + z2 j2 + z3 j6 + z4 j24 , (73)where z j = ∆t λ j and the λ j ∈ C are the N eigenvalues of A. The constantcoefficientstability constraint is|P (z j )| ≤ 1, ∀j ∈ [1, N]. (74)Alternatively, for all j, let z j = x j + iy j with x j , y j ∈ R, then the amplificationfactor g j of the fully discrete scheme has real partRe g j =()1 + x j + x2 j2 + x3 j6 + x4 j24− y2 j2(1 + x j + x2 j2)+ y4 j24(75)and imaginary partIm g j = y j()1 + x j + x2 j2 + x3 j6− y3 j6 (1 + x j) , (76)and our notion of stability implies that |g j | ≤ 1 for all j. If one can estimatethe eigenvalues of the spatial operator λ j , one then has a means of selecting astable timestep. In practice, the semi-ellipse shown in Figure 6,( ) 4x 2 ( ) 2y 2+ ≤ 1 for x ≤ 0, (77)11 5provides a more practical relation to determine stability.Analytically, the constant-coefficient problem reveals a potential shortcomingof our full discretization. Define the shift operator and its inverseT d u i = u i+e d and T −1d u i = u i−e d.The semi-discrete system of ordinary differential equations (61) reduces tod(uJ) idt= − 1 h( D ∑d=1a d[ 23( )Td − Td−1 1 −12( ) ])T2d − Td−2 (uJ) i. (78)On a periodic domain, the eigenvalues areλ i = − i3hD∑a d sin θ id [4 − cos θ id ] , (79)d=122