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# Adaptive order of block backward differentiation formulas for stiff ODEs

• In this paper, Adapative Order of Block Backward Differentiation Formulas (ABBDFs) are formulated using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs). These ABBDF methods are of order four, five and six. The benefit of the ABBDF methods is the computation time in the computation of solutions. Numerical results are presented to demonstrate the advantage of implementing adaptive order selection in a single code.

Mathematics Subject Classification: Primary: 65L04, 65L05; Secondary: 65L20.

 Citation: • • Figure 1.  Stability region of BBDF order 4, 5 and 6

Figure 2.  k-order ABBDF method

Figure 3.  Problem 3.1

Figure 4.  Problem 3.2

Table 1.  Coefficients for ABBDF

 $k$ $y_{n-4}$ $y_{n-3}$ $y_{n-2}$ $y_{n-1}$ $y_{n}$ $y_{n+1}$ $y_{n+2}$ $f_{n+1}$ $f_{n+2}$ 4 0 0 $\frac{1}{10}$ $-\frac{3}{5}$ $\frac{9}{5}$ 1 $-\frac{3}{10}$ $\frac{12}{10}$ 0 0 0 $-\frac{3}{25}$ $\frac{16}{25}$ $-\frac{36}{25}$ $\frac{48}{25}$ 1 0 $\frac{12}{25}$ 5 0 $-\frac{3}{65}$ $\frac{4}{13}$ $-\frac{12}{13}$ $\frac{24}{13}$ 1 $\frac{12}{65}$ $\frac{12}{13}$ 0 0 $\frac{12}{137}$ $-\frac{75}{137}$ $\frac{200}{137}$ $-\frac{300}{137}$ $\frac{300}{137}$ 1 0 $\frac{16}{137}$ 6 $\frac{2}{77}$ -$\frac{15}{77}$ $\frac{50}{77}$ $-\frac{100}{77}$ $\frac{150}{77}$ 1 $-\frac{10}{77}$ $\frac{60}{77}$ 0 -$\frac{10}{147}$ $\frac{24}{49}$ $-\frac{75}{49}$ $\frac{400}{147}$ $-\frac{150}{49}$ $\frac{120}{49}$ 1 0 $\frac{60}{147}$

Table 2.  Comparison between the ABBDF and BBDF for solving Problem 3.1

 $H$ Method MAXE TIME $10^{-2}$ BBDF(4) 5.34238e-6 0.008280686 BBDF(5) 8.60443e-6 0.008490485 BBDF(6) 1.09196e-5 0.0089108066 ABBDF 6.29854e-6 0.0082606264 $10^{-4}$ BBDF(4) 7.07097e-12 0.0145041159 BBDF(5) 3.99341e-11 0.01477376 BBDF(6) 5.75778e-11 0.01497369 ABBDF 2.51638e-11 0.014267131 $10^{-6}$ BBDF(4) 1.19693e-12 0.331891367 BBDF(5) 1.27653e-12 0.35464837 BBDF(6) 1.54599e-12 0.366591181 ABBDF 1.32749e-12 0.3299601339 $10^{-8}$ BBDF(4) 5.70600e-11 32.31383741 BBDF(5) 6.27814e-11 33.986499 BBDF(6) 8.38467e-11 35.61432122 ABBDF 6.87739e-11 32.222682490

Table 3.  Comparison between the ABBDF and BBDF for solving Problem 3.2

 $H$ Method MAXE TIME $10^{-2}$ BBDF(4) 6.13882e+1 0.008227781 BBDF(5) 2.14167e-1 0.0082600851 BBDF(6) 2.96326e+0 0.00828741 ABBDF 1.44159e-2 0.008064681 $10^{-4}$ BBDF(4) 7.76160e-8 0.01543074 BBDF(5) 9.48795e-8 0.016383168 BBDF(6) 1.11123e-7 0.0171090074 ABBDF 7.76160e-8 0.0142214673 $10^{-6}$ BBDF(4) 1.93315e-11 2.97284740 BBDF(5) 1.07853e-11 3.12728618 BBDF(6) 2.66430e-11 3.8798845185 ABBDF 1.93315e-11 2.31853289 $10^{-8}$ BBDF(4) 2.35567e-9 84.36481221 BBDF(5) 4.41047e-10 85.76521578 BBDF(6) 2.59949e-9 87.69974205 ABBDF 2.35567e-9 84.36227574

Table 4.  Comparison between the ABBDF and mathod in  for solving Problem 3.3

 $H$ Method MAXE $10^{-3}$ BBDF(5) 7.43088e-3 BDF 4.97642e+43 ode15s 6.30000e-3 ABBDF 2.02095e-3 $10^{-5}$ BBDF(5) 7.68218e-4 BDF 7.88641e-4 ode15s 6.40000e-3 ABBDF 6.22269e-4 $10^{-7}$ BBDF(5) 8.09607e-8 BDF 8.39232e-8 ode15s 4.30000e-5 ABBDF 6.49214e-8
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