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I have been using NDSolve, and I noticed that when you decrease the MaxStepSize, possible singularities are resolved.

If I decrease the step size past a certain point (less than 0.01), Wolfram Engine terminates because the peak system memory limit was reached.

Will a good computer + Mathematica do much better than a chromebook + Mathematica Online?

A = 1.4; B =20;
sol = NDSolve[{ D[v1[x, t], t] + 1/2 D[v1[x, t]^2, x] == (A )*v2[x, t] -  B v1[x, t],
D[v2[x, t], t] + 1/2 D[v2[x, t]^2, x] == (A)*v1[x, t] - B v2[x, t],
v1[x, 0] == 2, v2[x,0]==40, v2[0, t] == 40, v1[0,t] ==2, v2[10,t]==40},
{v1, v2}, {x, 0, 10}, {t, 0, 20}, MaxStepSize -> 0.001];
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    $\begingroup$ …you might want to consider posting the differential equation that is giving you trouble. $\endgroup$
    – J. M.'s torpor
    Jun 6 '15 at 5:32
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    $\begingroup$ In general, the maximum amount of memory for a cloud session is limited (0.5, 1 or 2 GB depending on the plan). If you need to use more, a computer with an appropriate amount of RAM will likely do better. $\endgroup$
    – ilian
    Jun 6 '15 at 5:39
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In general, the maximum amount of memory for a cloud session is limited (0.5, 1 or 2 GB depending on the plan). If you need to use more, a computer with an appropriate amount of RAM will likely do better.

– ilian Jun 6 at 5:39

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