... which of various alternative numericalmethods should be used for a specific problem, or even for a large class of problems 56 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS Table 201(II) h ... 12 12 −9 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS 0.05 h 0.10 0.15 62 Figure 203(iii) x Stepsize h against x for the ‘mildly stiff’ problem (203a) with variable stepsize for T = 0.02 ... function, y, on [x0 , x] by the formula y(x) = y(xk−1 ) + (x − xk−1 )f (xk−1 , y(xk−1 )), x ∈ (xk−1 , xk ], (210b) 66 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONSfor k = 1, 2, , n If we...
... converge for large stepsizes (not shown in the diagrams) This effect persisted for a larger range of stepsizes for PEC methods than was the case for PECE methods NUMERICAL METHODSFORORDINARYDIFFERENTIAL ... than for corresponding explicit NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS 10−6 10−4 104 −8 E 10 10 −10 10−4 Figure 239(ii) 10−3 h 10−2 Runge–Kutta methods with cost corrections methods ... with Figure 252(i) shows the new methods to be slightly more accurate for the same stepsizes 122 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS The final numerical result in this subsection...
... of the matrix A For i corresponding to a member of row k for k = 1, 2, , m, the only non-zero 190 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS aij are for j = and for j corresponding ... ) For explicit methods, D(2) cannot hold, for similar reasons to the impossibility of C(2) For implicit methods D(s) is possible, as we shall see in Section 342 174 NUMERICALMETHODSFORORDINARY ... 31.3 For an arbitrary Runge–Kutta method, find the order condition corresponding to the tree 170 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS 32 Low Order Explicit Methods 320 Methods...
... p + (333g) 204 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS Proof For a given tree t, let Φ(t) denote the elementary weight for (333a) and Φ(t) the elementary weight for (333b) Because ... I formula, c1 = This formula is exact for polynomials of degree up to 2s − II For the Radau II formula, cs = This formula is exact for polynomials of degree up to 2s − III For the Lobatto formula, ... of degree 216 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS ∗ less than n − A simple calculation shows that Q is orthogonal to Pk for ∗ k < n − Hence, (342f) follows except for the value...
... 230 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS 35 Stability of Implicit Runge–Kutta Methods 350 A-stability, A(α)-stability and L-stability We recall that the stability function for ... choose Z = −t diag(ej ), for t positive The value of R(Z) becomes R(Z) = − tbj + O(t2 ), 248 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS which is greater than for t sufficiently small Now ... non-linear equations We consider how to solve these equations using a full Newton method This requires going through the following steps: 260 NUMERICALMETHODSFORORDINARYDIFFERENTIAL EQUATIONS...
... Runge–Kutta methods exist for which A is lower triangular? 280 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS 38 Algebraic Properties of Runge–Kutta Methods 380 Motivation For any specific ... then the sub-forest induced by V is the forest (V , E), where E is the intersection of V × V and E A special 288 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS case is when a sub-forest (V ... acceptable for many problems In each of the values of ξ for which there is a single underline, the method is A(α)-stable with α ≥ 1.55 ≈ 89◦ 268 NUMERICALMETHODSFORORDINARYDIFFERENTIAL EQUATIONS...
... of this test in Subsection 433 346 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS Algorithm 432α Boundary locus method for low order Adams–Bashforth methods % Second order % -w = ... 348 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS 2i −6 −4 −2 −2i Figure 432(iii) Stability region for the third order Adams–Moulton method 2i −2i Figure 432(iv) Stability region for ... following equationsfor the predicted and corrected values: ∗ ∗ ∗ yn = yn−1 + hfn−1 − hfn−2 , (434a) 2 ∗ ∗ (434b) yn = yn−1 + hfn + hfn−1 2 350 NUMERICALMETHODSFORORDINARYDIFFERENTIAL EQUATIONS...
... form given by Exercise 53.1 420 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS 54 Methods with Runge–Kutta stability 540 Design criteria for general linear methods We consider some of the ... NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS stability properties that are usually superior to those of alternative methodsFor example, A-stability is inconsistent with high order for ... V as a simple matrix, for example a matrix with rank 422 NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS If p = q, it is a simple matter to write down conditions for this order and stage...
... relevant for computer methods, Chapter introduces all the basic concepts and simple methods (relevant also for boundary value problems and for DAEs), Chapter is devoted to one-step (Runge-Kutta) methods ... solutions for Example 9.9 252 9.4 A matrix in Hessenberg form 258 10.1 Methodsfor the direct discretization of DAEs in general form 265 10.2 Maximum errors for ... path for (x2 y2) 290 ; Chapter Ordinary Di erential EquationsOrdinary di erential equations (ODEs) arise in many instances when using mathematical modeling techniques for describing...
... Semigroups Parabolic Equations V Implicit Evolution Equations Introduction Regular Equations Pseudoparabolic Equations Degenerate Equations Examples ... so that j C0 j (x) for all x Rn , supp( j ) Gj , and j (x) = for x Fj LetS C0 (Rn ) be chosen with (x) n , supp( ) G for all x R fFj : j N g, and (x) P for x G = Finally, for each j , j N , ... Approximation of Evolution Equations Introduction Regular Equations Sobolev Equations Degenerate Equations Examples ...