Example 23 Second order integrator

[bslazarov]

Does anybody know the explicit form of the solution at step n+1 in example 23? I can see that d2udt2 = f(u + fac0d2udt2,dudt + fac1d2udt2, t); however, I am not sure why the solution at n+1, i.e., u+fac0*d2udt2, has contribution only from the acceleration term and does not include the velocity term.

[tzanio]

ping: @IdoAkkerman

[IdoAkkerman]

Short answer: Both the u and dudt variables in the Mult and ImplicitSolve should already be modified to include the know components of dudt and d2udt2. This should be taken care of in the ODE solver step. Longer answer: What we solve for is, f(u^a0, dudt^a1, d2udt2^a2) =0 Where a0, a1 and a2 are effectively timelevels (usually between n and n+1). To arrive at a solvable system interpolation (extrapolation) rules relating u, dudt and d2udt2 and different timelevels need to be provided. For a large part these defines the actual time integrator method used. This can be rewritten as f(c0 u^n + c1 dudt^n + c2 d2udt2^n + fac0 d2udt2^a2, c2 dudt^n + c3 d2udt2^n + fac1 d2udt2^a2, c4 d2udt2^n + fac1 d2udt2^a2) In the mult/implicitSolve routines we have the following association d2udt2 —> d2udt2^a2 this the unknown acceleration (at the intermediate level) for which we solve(march) Furthermore u = c0 u^n + c1 dudt^n + c2 d2udt2^n dudt = c0 u^n + c1 dudt^n + c2 d2udt2^n These are the “known" parts of unknown u^a0 and dudt^a1.

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On 13 Apr 2024, at 23:18, Boyan Lazarov ***@***.***> wrote: Does anybody know the explicit form of the solution at step n+1 in example 23? I can see that d2udt2 = f(u + fac0d2udt2,dudt + fac1d2udt2, t); however, I am not sure why the solution at n+1, i.e., u+fac0*d2udt2, has contribution only from the acceleration term and does not include the velocity term. — Reply to this email directly, view it on GitHub <#4246>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AAHVJ4342DXM7X3ILKEF6UDY5GOJJAVCNFSM6AAAAABGFYDOEWVHI2DSMVQWIX3LMV43ERDJONRXK43TNFXW4OZWGUYDANBRGU>. You are receiving this because you are subscribed to this thread.

[IdoAkkerman]

@4233 should fix the issue

[IdoAkkerman]

#4233