[Playlisten] [Impressum und Datenschutzerklärung]

Error of the Forward Euler Method, LTE


CC-BY-NC-SA 3.0

Tempo:

Anklickbares Transkript:

pronoun – want to look at the arrow that the forward on the method makes it each step – this is called the local – truncation – arrow critical dextran – rather – straightforward – to be looking at it again I'm considering – the differential equations of classical mechanics – for simplicity – just in pronoun dimension – the – derivative – of the position with respect to time is the city – derivative – of the velocity – with respect to time is – for us depending on position – I'm not looking at for spending – on velocities – are directly depending on time – easily incorporate that – all the house – this comes from Newton saying that foresees – response times acceleration – and acceleration – is the derivative of velocity with respect to time – if you solve this numeric key in the infection – this is precisely what the forward all the methods – which do – doing pronoun forward on the step – defines the time by slight – increments – by step size – pronoun want to compute – the position – after that step – this call that X two – from the position befallen standards call that X number – is – on equal there's some slight – increments – in this step size – H times the velocity – start – this is what will assess what we should be doing – take the original – ignition position – and involves that position by the time that's causing – during the step times the velocity at the beginning – similarly – for velocity – philosophy after that step is – for this – loss – step size times the rate of change of velocity which is the force – house – at – X1 – again be taking the force – at the start of – the visible fee would be computing – this is what should be doing – X X solving – determiner differential equation – would be actually shooting with the forward on this – tons of steps – of this sort Monson the position – on scene the arrow or that I'm making here – is called the local – truncation arrow – arrow – pronoun single step – is called the ACE – all – the local – truncation – arrow – and – whether – the question – how large or how small is that Al Gore – pronoun want to have an idea – about the arrow that I'm introducing – do pronoun have to use determiner smaller step size can pronoun get away with the largest of sizes – of dust that depend on the Mas on the force – on my carbon velocity – whatever – you can figure that out – to be saying that because I'm lazy – I'm working in pronoun dimension – there's determiner slight – change – to be – taken care of working more than pronoun dimension – pronoun mentioned that as pronoun go – that's it is red curve is the exact – solution – of my differential equation – the function of about approximate – using – numerics – using default on – do – know – the position – at – the start – of my step – time T one – this – would be – excellent – pronoun want to know what the position – is the end of that step – time T to – this is what pronoun would like to know – this is not X to X two was on numerical approximation – this will be – acts – of time – T to – the exact solution is twenty two the friends of T two and T one – step size stage – and all we can try and come up with some – approximations – for this dotted line – the value of X this time T to – first of all what RTM – would simply be excellent – the value that we started with – and then pronoun can make use of the slow – curve – if I'm at the tensions – so – say this new line is the tensions – to the rest exact curve – as time – T one – than the next – best approximation – for this video – the X pronoun class – this – ruling – year – and to get that from the derivative – you know the slope of the tangent – the slope of the tangent – is the derivative – of X – we know – this lack of this right triangle – that stage – so – the site has determiner link off – derivative – ST number – times – age – and he can do even better – instead of using the line – we can use determiner parabola – then the next – addition – to all approximation – the second – derivative – function – ST number – times age script over to – this man look really crazy why should that be determiner scratch over to – if you look at these you would be saying the next logical thing to be doing would be asked – second derivative – times age script – why should we be including this to – the ideas the following pronoun want to construct determiner parabola – that has the same where you seems – low in the same second derivative – is the original – red curve has asked T one – this component ensures that pronoun have the right ways you equal – to zero – pronoun got the right value for the – this component ensures that pronoun have the right slope – of the first derivative – expects – to H – Sage equal to zero – right – slow – and this time – ensures – that God – right coverture so to speak – if you found the second derivative of that star – you have to have the second – derivative – of that risk of ST number five the second derivative – age script – first regards to age and then becomes to – those two tools Kansas – so this too stems from forming the second derivative of age script with respect to age – this construction – is called Taylor expansion – in what ways does the following – the exact – value – of X – ST to – after the step – is – X number – plus what you get from the first derivative – dots – of T one – times step size class will get determiner second derivative – derivative – of X with respect to time at the start of the stick times – age script over two – plus – it's easy to imagine if you continue that to get something to age to the power and so on – we see that under the rock and say – order of – age to the power – if you want to get out this is called London oh oh symbol – so pronoun got determiner nice expression for the position after the step – there are some – things in here that you don't know yet – actually – do know – the first derivative – X with respect to time start is of course nothing but the velocity was starting – the second – derivative – of X with respect to time – at the start – is the first derivative – of the velocity – the first derivative of the velocity – is the force – that points overran – the force at that initial point X of T one – or – so that looks pretty handy – this is something we can compute – you do not know much about this – auto face to the power to you – but all of these can be computed – the initial position – initial velocity – step size – force – at the initial position – over the mosque times step size script over to all of these can be computed – we can do the same type of Taylor expansion – for the velocity – is pretty easy to see how that looks like the exact value of the velocity – should be the better you – before – the step – Sosnoff – causes the derivative – of the velocity – at the start – times the step size – plus the second – derivative – of the velocity – at the start times step size creates over two plus – again something of the order of age to our – the derivative – of the velocity with respect to time – that's the acceleration – we do know that – for so comma so this is the force – at the initial position – of the mouse – this pronoun here is determiner little tricky – second – derivative – of the velocity – with respect to time – to – do this pronoun more closely – the second derivative – of velocity with respect to time that provided like that – first the relative of the velocity – and then pronoun found the derivative – of that derivative – pronoun do know something about the first derivative – that's the acceleration – and pronoun know what acceleration – is the force depending on the position – must – analyze – the function – force – the defense on another function – the position that depends on time – this cries outlawed for the chain – first found out derivative – false over Moss – so – preventive – for – us – times – in the derivative – X thoughts – derivative – of X with respect to T – and we do know what next thought is simply the velocity – is it working in more than pronoun dimension this gets tricky – is not just the space recorded – X – there's also determiner why – do you see – so you have to follow several derivatives – here off the force vector – this ends up being determiner metrics – and this would be the product of metrics – times of vector – another – caveat – if the force depends on time directly – not just to the position – but if there's some – oscillating – pie for instance – the force – would have to have – not – the component in here in our situation does all got – which means that this Tom – question mark – John eat foods – spatial – derivative – of the force – at the initial position – Moss – times – the initial velocity – so also this can be computed from the data regarding our – simulation – we can look at those two equations – and try to decipher – what their meaning is X1 – plus – three number times age – this is pronoun – forward on the step four X – the pronoun plus the force at the start divided by him – this is the forward are listed so what you got your – is the forward on this – will be cut here on the left-hand side is the exact resides – so this remainder – must be the our – this is the local truncation era – of the forward are limited – to you can put numbers to that – given – the force given the mosque – and given the – spatial – derivative of the force – the initial velocity – and so on you can – get determiner pretty good estimate – of the local truncation – before all of you see – that this is of the order H Square – and if you look closely – you see determiner way to improve the forward are limited – let's use what we know from the forward on the method but – this pronoun the force at the initial position – over Moss times determiner script over to – and similar here that pretty – complicated – term in the ends – then he got determiner method that has determiner local truncation – of audit H to the power – this is called – Taylor serious – to make this tea is off to – of course – this improvement – doesn't come for free – you have to compute the spatial derivative – force