A Random Choice Finite Difference Scheme for Hyperbolic Conservation
A Random Choice Finite Difference Scheme for Hyperbolic Conservation
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27) gives r^ da < 0(A) [NA + (NA) ^] Now by (3. 5b), NA = NtX = TA = 0(1); it follows that (3. 29) I r^ da £ 0(A) . Denote by m(e) the a-measure of the set where |r(a) [ > e . 2 1/3 It follows from (3. 29) that e m(e) £ 0(A). Taking e = A ^ we deduce 1/3 Lemma 3. 3 . |r(a) | f. A ^ for all except on a set 1/3 of measure <_ 0(A ) . We deal with entropy in an analogous manner. The integral form of the entropy condition (1. 8b) is that (3. 30) - I I Lw^U + w^F] dx dt - w(x, 0) U(x, 0) dx < for all smooth, nonnegative test functions w(x, t). The discrete version of (3. 30), after summing by parts, is 27 N (3. 31) R = I R where (3. 32) ^n = ] w(x, t^) [U(v(x, tj^_^^) - U(v(x, t^) + tD^F (v (x, t^) ] dx . The analogue of Lemma 3. 2 is Lemma 3. 4 . (i) Ir (a) I < 0(A) for all a (3. 33) (ii) I R da < O(A^) n — The proof of (i) is the same as before; the proof of (ii) is analogous; instead of estimates for the absolute value of M we have only upper bounds for M, and correspondingly only an upper bound in (3.
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To quickly assess the difficulty of the text, read a short excerpt:
27) gives r^ da < 0(A) [NA + (NA) ^] Now by (3. 5b), NA = NtX = TA = 0(1); it follows that (3. 29) I r^ da £ 0(A) . Denote by m(e) the a-measure of the set where |r(a) [ > e . 2 1/3 It follows from (3. 29) that e m(e) £ 0(A). Taking e = A ^ we deduce 1/3 Lemma 3. 3 . |r(a) | f. A ^ for all except on a set 1/3 of measure <_ 0(A ) . We deal with entropy in an analogous manner. The integral form of the entropy condition (1. 8b) is that (3. 30) - I I Lw^U + w^F] dx dt - w(x, 0) U(x, 0) dx < for all smooth, nonnegative test functions w(x, t). The discrete version of (3. 30), after summing by parts, is 27 N (3. 31) R = I R where (3. 32) ^n = ] w(x, t^) [U(v(x, tj^_^^) - U(v(x, t^) + tD^F (v (x, t^) ] dx . The analogue of Lemma 3. 2 is Lemma 3. 4 . (i) Ir (a) I < 0(A) for all a (3. 33) (ii) I R da < O(A^) n — The proof of (i) is the same as before; the proof of (ii) is analogous; instead of estimates for the absolute value of M we have only upper bounds for M, and correspondingly only an upper bound in (3.
What to read after A Random Choice Finite Difference Scheme for Hyperbolic Conservation?
You can find similar books in the "Read Also" column, or choose other free books by A Harten to read online
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