[linux-lvm] Replace small drive with bigger, using LVM?

[Craig]

I have a workstation that can only take 2 hard drives.
The one that's in there is a 500GB drive and I'd like to bring a new 4TB 
drive in and use LVMto migrate a live running system over to the new 
drive, remove the old one from the setup, reboot and bing, have the 
large drive running,without having to re-install my OS and set up 
everything afresh.
It's an Ubuntu 14.04.1 install where I ticked the "Use LVM" tick box

# fdisk -l

Disk /dev/sda: 500.1 GB, 500107862016 bytes
255 heads, 63 sectors/track, 60801 cylinders, total 976773168 sectors
Units = sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0x0005d771

    Device Boot      Start         End      Blocks   Id  System
/dev/sda1   *        2048      499711      248832   83  Linux
/dev/sda2          501758   976771071   488134657    5  Extended
/dev/sda5          501760   976771071   488134656   8e  Linux LVM

Disk /dev/sdb: 4000.8 GB, 4000787030016 bytes
255 heads, 63 sectors/track, 486401 cylinders, total 7814037168 sectors
Units = sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 4096 bytes
I/O size (minimum/optimal): 4096 bytes / 4096 bytes
Disk identifier: 0xc73f6184

    Device Boot      Start         End      Blocks   Id  System
/dev/sdb1            2048      499711      248832   83  Linux
/dev/sdb2          499712  4294967294  2147233791+   5  Extended
/dev/sdb5          501760  4294967294  2147232767+  8e  Linux LVM

Disk /dev/mapper/ubuntu--vg-root: 493.4 GB, 493363396608 bytes
255 heads, 63 sectors/track, 59981 cylinders, total 963600384 sectors
Units = sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0x00000000

Disk /dev/mapper/ubuntu--vg-root doesn't contain a valid partition table

Disk /dev/mapper/ubuntu--vg-swap_1: 6438 MB, 6438256640 bytes
255 heads, 63 sectors/track, 782 cylinders, total 12574720 sectors
Units = sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0x00000000

Disk /dev/mapper/ubuntu--vg-swap_1 doesn't contain a valid partition table

So /dev/sda is the original install disk and /dev/sdb in my shiny new WD 
Black 4TB drive.

# vgdisplay
   --- Volume group ---
   VG Name               ubuntu-vg
   System ID
   Format                lvm2
   Metadata Areas        1
   Metadata Sequence No  7
   VG Access             read/write
   VG Status             resizable
   MAX LV                0
   Cur LV                2
   Open LV               2
   Max PV                0
   Cur PV                1
   Act PV                1
   VG Size               465.52 GiB
   PE Size               4.00 MiB
   Total PE              119173
   Alloc PE / Size       119162 / 465.48 GiB
   Free  PE / Size       11 / 44.00 MiB
   VG UUID               tFSX8N-c3zG-j9BW-yqx2-Or63-AXMg-Sho5tD

root at home:~# lvdisplay
   --- Logical volume ---
   LV Path                /dev/ubuntu-vg/root
   LV Name                root
   VG Name                ubuntu-vg
   LV UUID                rqvzFC-b1I2-sLRm-oq6w-fTEP-UGSe-7E7fwz
   LV Write Access        read/write
   LV Creation host, time ubuntu, 2014-08-15 14:24:31 +0100
   LV Status              available
   # open                 1
   LV Size                459.48 GiB
   Current LE             117627
   Segments               2
   Allocation             inherit
   Read ahead sectors     auto
   - currently set to     256
   Block device           252:0

   --- Logical volume ---
   LV Path                /dev/ubuntu-vg/swap_1
   LV Name                swap_1
   VG Name                ubuntu-vg
   LV UUID                y9fXgx-ceMm-lss0-ZZi8-d2us-KdGT-ipUQQ4
   LV Write Access        read/write
   LV Creation host, time ubuntu, 2014-08-15 14:24:31 +0100
   LV Status              available
   # open                 2
   LV Size                6.00 GiB
   Current LE             1535
   Segments               1
   Allocation             inherit
   Read ahead sectors     auto
   - currently set to     256
   Block device           252:1


So I've got /boot in a primary partition and then swap and / come via 
LVM in the extended.

So I'd love to do something like
* dd /dev/sda1 to /dev/sdb1 or something like that and have my /boot 
volume set up
* install grub into the MBR of /dev/sdb

The if I can make the right LVM setup, get / from my live system copied 
to the new drive, remove the old drive lv from the vg and then 
physically swap the HDDs round.

Can someone help me with the finer details of this or isn't it doable?
TIA
-C