... why we will consider numericalmethodsfor solving this kind of differential equations However, the exact solution in hand can be used as benchmark to evaluate numericalmethods In the following ... Please refer to [12] for more details Other than finite element method, in this paper, we will discuss an alternative numerical method to handle the equations with distributions For the purpose of ... this thesis, we will mainly consider the numerical method for differential equations with the delta distribution and its distributional derivatives Such equations have very strong physical backgrounds...
... which of various alternative numericalmethods should be used for a specific problem, or even for a large class of problems 56 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS Table 201(II) h ... 12 12 −9 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS 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 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONSfor k = 1, 2, , n If we...
... to 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 METHODSFOR ORDINARY ... than for corresponding explicit NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS 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 ... criteria to derive Adams–Bashforth methods with p = k for k = 2, 3, 4, and Adams–Moulton methods with p = k + for k = 1, 2, For k = 4, the Taylor expansion of (241c) takes the form hy (xn )(1 − β0 −...
... 1/γ(t3 ) 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 NUMERICALMETHODSFOR ORDINARY ... of the matrix A For i corresponding to a member of row k for k = 1, 2, , m, the only non-zero 190 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS aij are for j = and for j corresponding ... 31.3 For an arbitrary Runge–Kutta method, find the order condition corresponding to the tree 170 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS 32 Low Order Explicit Methods 320 Methods...
... 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, ... p + (333g) 204 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS Proof For a given tree t, let Φ(t) denote the elementary weight for (333a) and Φ(t) the elementary weight for (333b) Because ... of degree 216 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS ∗ less than n − A simple calculation shows that Q is orthogonal to Pk for ∗ k < n − Hence, (342f) follows except for the value...
... 12 36 For E(y) ≥ 0, for all y > 0, it is necessary and sufficient for A-stability that λ ∈ [ , λ], where λ ≈ 1.0685790213 is a zero of the coefficient of y in E(y) For 262 NUMERICALMETHODSFOR ORDINARY ... choose Z = −t diag(ej ), for t positive The value of R(Z) becomes R(Z) = − tbj + O(t2 ), 248 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS which is greater than for t sufficiently small Now ... the form G u v , G u v = u v ≤ 0, where G is defined by f (u) f (v) Furthermore, the requirement on a numerical method (357b) can be written in the form Yn ≤ Yn−1 250 NUMERICALMETHODSFOR ORDINARY...
... Runge–Kutta methods exist for which A is lower triangular? 280 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS 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 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS case is when a sub-forest (V ... aij + bj aji − bi bj (370a) Now consider a problem for which y Qf (y) = 0, (370b) 276 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONSfor all y It is assumed that Q is a symmetric matrix...
... of this test in Subsection 433 346 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS Algorithm 432α Boundary locus method for low order Adams–Bashforth methods % Second order % -w = ... variable order formulation It is natural to make a comparison between implementation techniques for Runge–Kutta methods and for linear multistep methods Unlike for explicit Runge–Kutta methods, interpolation ... 348 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS 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...
... form given by Exercise 53.1 420 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS 54 Methods with Runge–Kutta stability 540 Design criteria for general linear methods We consider some of the ... NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS 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 NUMERICALMETHODSFOR ORDINARY DIFFERENTIAL EQUATIONS If p = q, it is a simple matter to write down conditions for this order and stage...
... 65 Numericalmethods 66 3.6.1 3.6.2 3.7 Numerical method for the quasi-2D equation I 67 Numerical method for the quasi-1D equation 69 Numerical ... parameter regimes Numericalmethods are proposed to compute the ground states for reduced equations 3.1 Lower dimensional models for dipolar GPE For the 3D dipolar GPE (2.5) which is reformulated into ... between different numericalmethodsfor GPE, or in a more general case, for the nonlinear Schr¨dinger equation (NLSE), we refer to [25, 47, 105, 144] o and references therein For ground states,...